Subgroups — #30858418

Order 17152.

1000 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863414	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863412	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863378	C₂	1	1	yes	maximal, characteristic
5	#22568627	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560876	C₂	1	8	yes	maximal, complemented
7	#22559918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11761628	C₂²	1	1	yes	characteristic
10	C₂²×#11761612	C₂²	1	1	yes	characteristic
11	C₂²×#11761612	C₂²	1	1	yes	characteristic
12	C₂×#14524002	C₂²	1	1	yes	characteristic
13	#16863444	—	2	2	no	—
14	#16863444	C₂²	1	4	yes	complemented
15	C₂×#14523978	—	2	2	no	—
16	#16863414	—	2	2	no	—
17	#16863414	C₂²	1	4	yes	—
18	#16863412	—	2	2	no	—
19	#16863412	—	2	2	no	—
20	C₂×#14523971	—	2	2	no	—
21	C₂×#14523969	C₂²	1	4	yes	—
22	#16863378	C₄	1	4	yes	complemented
23	#16863378	C₂²	1	4	yes	—
24	C₂×#14523961	C₄	1	2	yes	complemented
25	C₂×#14523959	C₄	1	2	yes	complemented
26	#16863354	—	2	2	no	—
27	#16863351	—	2	2	no	—
28	M₁₆×Dic₆₇	C₂²	1	4	yes	complemented
29	#16863155	C₂²	1	4	yes	complemented
30	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
33	#344×C₆₇	C₂²	1	4	yes	complemented
34	C₂×#11761628	—	2	1	no	—
35	C₂×#11761628	—	2	1	no	—
36	C₂×#11761628	—	2	1	no	—
37	C₂×#11761628	—	2	1	no	—
38	C₂×#11761628	—	2	1	no	—
39	C₂×#11761628	—	2	1	no	—
40	C₂×#11761628	—	2	1	no	—
41	C₂×#11761628	—	2	1	no	—
42	C₂×#11761628	—	2	1	no	—
43	C₂×#11761628	—	2	1	no	—
44	C₂×#11761628	C₂×C₄	1	2	yes	—
45	C₂×#11761628	C₂³	1	2	yes	—
46	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₂²⋊D₆₇	—	2	1	no	—
49	C₂²×C₄×D₆₇	—	2	1	no	—
50	C₂×#11761612	—	2	1	no	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761612	—	2	1	no	—
53	C₂×#11761612	—	2	1	no	—
54	C₂×#11761612	—	2	1	no	—
55	C₂×#11761612	—	2	1	no	—
56	C₂×#11761612	C₂×C₄	1	2	yes	—
57	C₂×#11761612	C₂×C₄	1	2	yes	—
58	C₂×#11761612	D₄	1	2	yes	—
59	C₂×#11761612	D₄	1	2	yes	—
60	C₂×#11761612	C₂³	1	2	yes	—
61	C₂×#11761612	C₂³	1	2	yes	—
62	C₂²×C₆₇⋊C₈	—	2	1	no	—
63	C₂²×C₆₇⋊C₈	—	2	1	no	—
64	#14524002	—	2	1	no	—
65	#14524002	—	2	1	no	—
66	#14524002	D₄	1	2	yes	—
67	#14524002	D₄	1	2	yes	—
68	C₂×C₄⋊Dic₆₇	—	2	1	no	—
69	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
70	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
71	#14523978	—	2	8	no	—
72	#14523971	—	2	8	no	—
73	#14523969	—	2	8	no	—
74	#14523961	C₂×C₄	1	8	yes	complemented
75	#14523959	C₂×C₄	1	8	yes	complemented
76	#14523953	—	2	4	no	—
77	C₈×Dic₆₇	—	2	4	no	—
78	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×M₁₆×C₆₇	—	2	1	no	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	—	2	1	no	—
82	C₂×M₁₆×C₆₇	—	2	1	no	—
83	C₂×M₁₆×C₆₇	—	2	1	no	—
84	C₂×M₁₆×C₆₇	—	2	1	no	—
85	C₂×M₁₆×C₆₇	—	2	1	no	—
86	C₂×M₁₆×C₆₇	—	2	1	no	—
87	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
89	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
90	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂²×C₈×C₆₇	—	2	1	no	—
92	C₂²×C₈×C₆₇	—	2	1	no	—
93	#117×C₆₇	—	2	1	no	—
94	#117×C₆₇	—	2	1	no	—
95	#117×C₆₇	—	2	1	no	—
96	#117×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂³×D₆₇	—	2	1	no	—
101	#11761628	—	2	2	no	—
102	#11761628	—	2	2	no	—
103	#11761628	—	2	2	no	—
104	#11761628	—	2	2	no	—
105	#11761628	—	2	2	no	—
106	#11761628	—	2	2	no	—
107	#11761628	—	2	2	no	—
108	#11761628	—	2	2	no	—
109	#11761628	—	4	1	no	—
110	#11761628	—	4	1	no	—
111	#11761628	—	4	1	no	—
112	#11761628	—	4	1	no	—
113	#11761628	—	4	1	no	—
114	#11761628	—	4	1	no	—
115	#11761628	—	4	1	no	—
116	#11761628	—	4	1	no	—
117	C₂×Dic₁₃₄	—	2	1	no	—
118	C₂×Dic₁₃₄	—	2	1	no	—
119	C₂×Dic₁₃₄	—	2	1	no	—
120	C₂×Dic₁₃₄	—	2	1	no	—
121	C₂×Dic₁₃₄	—	2	1	no	—
122	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
123	C₂×D₂₆₈	—	2	1	no	—
124	C₂×D₂₆₈	—	2	1	no	—
125	C₂×D₂₆₈	—	2	1	no	—
126	C₂×D₂₆₈	—	2	1	no	—
127	C₂×D₂₆₈	—	2	1	no	—
128	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
129	C₂×C₂²⋊D₆₇	—	2	2	no	—
130	C₂×C₂²⋊D₆₇	—	2	2	no	—
131	C₂×C₂²⋊D₆₇	—	4	1	no	—
132	C₂×C₂²⋊D₆₇	—	4	1	no	—
133	C₂×C₂²⋊D₆₇	—	4	1	no	—
134	C₂×C₂²⋊D₆₇	—	4	1	no	—
135	C₂²×Dic₆₇	—	2	1	no	—
136	C₂²×Dic₆₇	—	2	2	no	—
137	C₂×C₄×D₆₇	—	2	2	no	—
138	C₂×C₄×D₆₇	—	2	2	no	—
139	C₂×C₄×D₆₇	—	4	1	no	—
140	C₂×C₄×D₆₇	—	4	1	no	—
141	C₂×C₄×D₆₇	—	4	1	no	—
142	C₂×C₄×D₆₇	—	4	1	no	—
143	#11761612	—	2	2	no	—
144	#11761612	—	2	2	no	—
145	#11761612	—	2	2	no	—
146	#11761612	—	2	2	no	—
147	#11761612	—	2	2	no	—
148	#11761612	—	2	2	no	—
149	#11761612	C₂²×C₄	1	4	yes	—
150	#11761612	C₂×D₄	1	4	yes	—
151	C₂×C₆₇⋊C₈	—	2	2	no	—
152	C₂×C₆₇⋊C₈	—	2	2	no	—
153	C₂×C₆₇⋊C₈	—	2	2	no	—
154	C₂×C₆₇⋊C₈	—	2	2	no	—
155	C₂×C₆₇⋊C₈	—	2	2	no	—
156	C₂×C₆₇⋊C₈	—	2	2	no	—
157	C₄⋊Dic₆₇	—	2	2	no	—
158	C₄⋊Dic₆₇	—	2	2	no	—
159	C₂²⋊Dic₆₇	—	2	2	no	—
160	C₂²⋊Dic₆₇	—	2	2	no	—
161	C₄×Dic₆₇	—	2	2	no	—
162	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
163	C₂⁴×C₆₇	#41	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	—	4	1	no	—
165	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	M₁₆×C₆₇	—	2	2	no	—
180	M₁₆×C₆₇	—	2	2	no	—
181	M₁₆×C₆₇	—	2	2	no	—
182	M₁₆×C₆₇	—	2	2	no	—
183	M₁₆×C₆₇	—	2	2	no	—
184	M₁₆×C₆₇	—	4	1	no	—
185	M₁₆×C₆₇	—	4	1	no	—
186	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
187	C₂×C₈×C₆₇	—	2	2	no	—
188	C₂×C₈×C₆₇	—	2	2	no	—
189	C₂×C₈×C₆₇	—	2	2	no	—
190	C₂×C₈×C₆₇	—	2	2	no	—
191	C₂×C₈×C₆₇	—	4	1	no	—
192	C₂×C₈×C₆₇	—	4	1	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	4	1	no	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	4	1	no	—
197	C₄²×C₆₇	—	2	2	no	—
198	C₄²×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	8	1	no	—
200	C₂³×C₆₇	#117	1	1	yes	characteristic
201	C₂³×C₆₇	#117	1	1	yes	characteristic
202	C₂³×C₆₇	#123	1	1	yes	characteristic
203	C₂³×C₆₇	#123	1	1	yes	characteristic
204	C₂³×C₆₇	#124	1	1	yes	characteristic
205	C₂³×C₆₇	#124	1	1	yes	characteristic
206	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₂²×D₆₇	—	2	2	no	—
208	C₂²×D₆₇	—	4	1	no	—
209	C₂²×D₆₇	—	4	1	no	—
210	C₂²×D₆₇	—	8	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	8	2	no	—
213	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
214	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
224	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
225	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
226	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
227	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
228	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
229	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
230	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
231	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
232	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
233	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
234	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
241	C₂²⋊D₆₇	—	4	2	no	—
242	C₂²⋊D₆₇	—	4	2	no	—
243	C₂²⋊D₆₇	—	4	2	no	—
244	C₂²⋊D₆₇	—	4	2	no	—
245	C₂×Dic₆₇	—	2	2	no	—
246	C₂×Dic₆₇	—	2	4	no	—
247	C₂×Dic₆₇	—	4	1	no	—
248	C₂×Dic₆₇	—	4	1	no	—
249	C₂×Dic₆₇	—	8	2	no	—
250	D₂₆₈	—	2	2	no	—
251	D₂₆₈	—	2	2	no	—
252	D₂₆₈	—	4	1	no	—
253	D₂₆₈	—	4	1	no	—
254	C₄×D₆₇	—	4	2	no	—
255	C₄×D₆₇	—	4	2	no	—
256	C₄×D₆₇	—	4	2	no	—
257	C₄×D₆₇	—	4	2	no	—
258	Dic₁₃₄	—	2	2	no	—
259	Dic₁₃₄	—	2	2	no	—
260	Dic₁₃₄	—	4	1	no	—
261	Dic₁₃₄	—	4	1	no	—
262	C₈×C₆₇	—	2	4	no	—
263	C₈×C₆₇	—	4	2	no	—
264	C₆₇⋊C₈	—	2	4	no	—
265	C₆₇⋊C₈	—	2	4	no	—
266	C₂²×C₆₇	—	4	1	no	—
267	C₂²×C₆₇	—	4	1	no	—
268	C₂²×C₆₇	—	4	1	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	1	no	—
271	C₂²×C₆₇	—	4	1	no	—
272	C₂²×C₆₇	—	4	1	no	—
273	C₂²×C₆₇	#388	1	1	yes	characteristic
274	C₂²×C₆₇	#388	1	1	yes	characteristic
275	C₂²×C₆₇	#388	1	1	yes	characteristic
276	C₂²×C₆₇	#388	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
280	C₂×D₆₇	—	4	2	no	—
281	C₂×D₆₇	—	8	2	no	—
282	C₂×D₆₇	—	8	2	no	—
283	C₂×D₆₇	—	8	2	no	—
284	C₄×C₆₇	—	8	2	no	—
285	C₄×C₆₇	#388	1	1	yes	characteristic
286	C₄×C₆₇	#388	1	1	yes	characteristic
287	C₄×C₆₇	#388	1	1	yes	characteristic
288	C₄×C₆₇	#388	1	1	yes	characteristic
289	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
290	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
291	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
292	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
293	Dic₆₇	—	4	4	no	—
294	Dic₆₇	—	4	2	no	—
295	#21080	—	67	1	no	maximal, Sylow
296	C₂×C₆₇	—	2	1	no	—
297	C₂×C₆₇	—	2	1	no	—
298	C₂×C₆₇	—	2	1	no	—
299	C₂×C₆₇	—	2	1	no	—
300	C₂×C₆₇	—	2	1	no	—
301	C₂×C₆₇	—	2	1	no	—
302	C₂×C₆₇	—	2	1	no	—
303	C₂×C₆₇	C₂×#388	1	1	yes	characteristic
304	D₆₇	—	8	2	no	—
305	C₂×#430	—	67	1	no	—
306	C₂×#421	—	67	1	no	—
307	C₂×#421	—	67	1	no	—
308	C₂×#413	—	67	1	no	—
309	#2873	—	67	1	no	—
310	#1932	—	67	8	no	—
311	C₂×#344	D₆₇	1	1	yes	characteristic, complemented
312	#1733	—	67	1	no	—
313	C₆₇	#21080	1	1	yes	characteristic, complemented, Sylow
314	C₂²×#41	—	67	1	no	—
315	C₂²×M₁₆	—	67	1	no	—
316	C₂²×M₁₆	—	67	1	no	—
317	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
318	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
319	C₂×#117	—	67	1	no	—
320	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
321	#430	—	67	4	no	—
322	#430	—	134	2	no	—
323	C₂×#108	—	134	2	no	—
324	#421	—	67	4	no	—
325	#421	—	134	2	no	—
326	#421	—	134	2	no	—
327	#421	—	134	2	no	—
328	C₂×#104	—	67	4	no	—
329	C₂×#104	—	134	2	no	—
330	#413	—	67	4	no	—
331	#413	—	67	4	no	—
332	C₂×#101	—	67	2	no	—
333	C₂×#100	—	67	2	no	—
334	#405	—	134	2	no	—
335	#405	—	134	2	no	—
336	C₄×M₁₆	—	67	4	no	—
337	#344	C₂×D₆₇	1	4	yes	complemented
338	#339	—	67	4	no	—
339	C₂×#41	—	67	2	no	—
340	C₂×#41	—	67	2	no	—
341	C₂×#41	—	134	1	no	—
342	C₂×#41	—	134	1	no	—
343	C₂×#41	—	134	1	no	—
344	C₂×#41	—	134	1	no	—
345	C₂×#41	—	134	1	no	—
346	C₂×#41	—	134	1	no	—
347	C₂×#41	—	134	1	no	—
348	C₂×#41	—	134	1	no	—
349	C₂×#41	—	134	1	no	—
350	C₂×#41	—	134	1	no	—
351	C₂²×Q₈	—	67	1	no	—
352	C₂²×D₄	—	67	1	no	—
353	C₂²×D₄	—	134	1	no	—
354	C₂³×C₄	—	134	1	no	—
355	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂×M₁₆	—	2	1	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂×M₁₆	—	2	1	no	—
364	C₂×M₁₆	—	67	2	no	—
365	C₂×M₁₆	—	67	2	no	—
366	C₂×M₁₆	—	67	2	no	—
367	C₂×M₁₆	—	67	2	no	—
368	C₂×M₁₆	—	67	2	no	—
369	C₂×M₁₆	—	67	2	no	—
370	C₂×M₁₆	—	134	1	no	—
371	C₂×M₁₆	—	134	1	no	—
372	C₂×M₁₆	—	134	1	no	—
373	C₂×M₁₆	—	134	1	no	—
374	C₂×M₁₆	—	134	1	no	—
375	C₂×M₁₆	—	134	1	no	—
376	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
377	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
378	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
379	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
380	C₂²×C₈	—	2	1	no	—
381	C₂²×C₈	—	2	1	no	—
382	C₂²×C₈	—	134	1	no	—
383	C₂²×C₈	—	134	1	no	—
384	#117	—	2	1	no	—
385	#117	—	2	1	no	—
386	#117	—	2	1	no	—
387	#117	—	67	2	no	—
388	#117	—	67	2	no	—
389	#117	—	134	1	no	—
390	#117	—	134	1	no	—
391	#117	C₂²×D₆₇	1	2	yes	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	134	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	134	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	67	2	no	—
398	#108	—	134	8	no	—
399	#104	—	134	8	no	—
400	#104	—	134	8	no	—
401	#101	—	67	8	no	—
402	#100	—	67	8	no	—
403	#97	—	134	4	no	—
404	C₄×C₈	—	134	4	no	—
405	C₂⁴	—	134	1	no	—
406	C₂⁴	#11761628	1	1	yes	characteristic
407	#41	—	268	1	no	—
408	#41	—	268	1	no	—
409	#41	—	268	1	no	—
410	#41	—	268	1	no	—
411	#41	—	268	1	no	—
412	#41	—	268	1	no	—
413	#41	—	268	1	no	—
414	#41	—	268	1	no	—
415	#41	—	134	2	no	—
416	#41	—	134	2	no	—
417	#41	—	134	2	no	—
418	#41	—	134	2	no	—
419	#41	—	134	2	no	—
420	#41	—	134	2	no	—
421	#41	—	134	2	no	—
422	#41	—	134	2	no	—
423	C₂×Q₈	—	67	2	no	—
424	C₂×Q₈	—	134	1	no	—
425	C₂×Q₈	—	134	1	no	—
426	C₂×Q₈	—	134	1	no	—
427	C₂×Q₈	—	134	1	no	—
428	C₂×Q₈	—	134	1	no	—
429	C₂×D₄	—	268	1	no	—
430	C₂×D₄	—	268	1	no	—
431	C₂×D₄	—	268	1	no	—
432	C₂×D₄	—	268	1	no	—
433	C₂×D₄	—	67	2	no	—
434	C₂×D₄	—	134	2	no	—
435	C₂×D₄	—	134	2	no	—
436	C₂×D₄	—	134	1	no	—
437	C₂×D₄	—	134	1	no	—
438	C₂×D₄	—	134	1	no	—
439	C₂×D₄	—	134	1	no	—
440	C₂×D₄	—	134	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	268	1	no	—
443	C₂²×C₄	—	268	1	no	—
444	C₂²×C₄	—	268	1	no	—
445	C₂²×C₄	—	268	1	no	—
446	C₂²×C₄	—	134	1	no	—
447	C₂²×C₄	—	134	2	no	—
448	C₂²×C₄	—	134	2	no	—
449	C₂²×C₄	—	134	2	no	—
450	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
452	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
453	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
454	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
455	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
456	C₂²×C₄	#11761627	1	1	yes	characteristic
457	C₂²×C₄	#11761627	1	1	yes	characteristic
458	C₂²×C₄	#11761628	1	1	yes	characteristic
459	C₂²×C₄	#11761628	1	1	yes	characteristic
460	C₂²×C₄	#11761628	1	1	yes	characteristic
461	C₂²×C₄	#11761629	1	1	yes	characteristic
462	C₂²×C₄	#11761629	1	1	yes	characteristic
463	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
464	M₁₆	—	2	2	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	2	2	no	—
467	M₁₆	—	2	2	no	—
468	M₁₆	—	2	2	no	—
469	M₁₆	—	4	1	no	—
470	M₁₆	—	4	1	no	—
471	M₁₆	—	67	4	no	—
472	M₁₆	—	67	4	no	—
473	M₁₆	—	134	2	no	—
474	M₁₆	—	134	2	no	—
475	M₁₆	—	134	2	no	—
476	M₁₆	—	134	2	no	—
477	M₁₆	—	134	2	no	—
478	M₁₆	—	134	2	no	—
479	M₁₆	C₂×C₄×D₆₇	1	4	yes	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	4	1	no	—
485	C₂×C₈	—	4	1	no	—
486	C₂×C₈	—	134	2	no	—
487	C₂×C₈	—	134	2	no	—
488	C₂×C₈	—	134	2	no	—
489	C₂×C₈	—	134	2	no	—
490	C₂×C₈	—	134	2	no	—
491	C₂×C₈	—	134	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₄⋊C₄	—	134	2	no	—
495	C₄⋊C₄	—	134	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	134	2	no	—
499	C₂²⋊C₄	—	134	2	no	—
500	C₄²	—	2	2	no	—
501	C₄²	—	4	1	no	—
502	C₄²	—	67	4	no	—
503	C₄²	—	134	2	no	—
504	C₂³	—	8	1	no	—
505	C₂³	—	268	1	no	—
506	C₂³	—	268	1	no	—
507	C₂³	—	536	2	no	—
508	C₂³	—	134	2	no	—
509	C₂³	#14524000	1	1	yes	characteristic
510	C₂³	#14524000	1	1	yes	characteristic
511	C₂³	#14524028	1	1	yes	characteristic
512	C₂³	#14524028	1	1	yes	characteristic
513	C₂³	#14524029	1	1	yes	characteristic
514	C₂³	#14524029	1	1	yes	characteristic
515	C₂³	C₂×#11761628	1	1	yes	characteristic
516	Q₈	—	268	1	no	—
517	Q₈	—	268	1	no	—
518	Q₈	—	134	2	no	—
519	Q₈	—	134	2	no	—
520	D₄	—	268	2	no	—
521	D₄	—	268	2	no	—
522	D₄	—	268	2	no	—
523	D₄	—	268	2	no	—
524	D₄	—	268	1	no	—
525	D₄	—	268	1	no	—
526	D₄	—	134	2	no	—
527	D₄	—	134	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	8	2	no	—
530	C₂×C₄	—	268	1	no	—
531	C₂×C₄	—	268	1	no	—
532	C₂×C₄	—	268	2	no	—
533	C₂×C₄	—	268	2	no	—
534	C₂×C₄	—	268	2	no	—
535	C₂×C₄	—	268	2	no	—
536	C₂×C₄	—	536	2	no	—
537	C₂×C₄	—	134	2	no	—
538	C₂×C₄	—	134	4	no	—
539	C₂×C₄	#14524000	1	1	yes	characteristic
540	C₂×C₄	#14524000	1	1	yes	characteristic
541	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
542	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
543	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
544	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
545	C₂×C₄	#14524005	1	1	yes	characteristic
546	C₂×C₄	#14524005	1	1	yes	characteristic
547	C₂×C₄	#14524005	1	1	yes	characteristic
548	C₂×C₄	#14524005	1	1	yes	characteristic
549	C₂×C₄	#14524016	1	1	yes	characteristic
550	C₂×C₄	#14524016	1	1	yes	characteristic
551	C₂×C₄	#14524016	1	1	yes	characteristic
552	C₂×C₄	#14524016	1	1	yes	characteristic
553	C₂×C₄	#14524028	1	1	yes	characteristic
554	C₂×C₄	#14524028	1	1	yes	characteristic
555	C₂×C₄	#14524029	1	1	yes	characteristic
556	C₂×C₄	#14524029	1	1	yes	characteristic
557	C₂×C₄	#14524038	1	1	yes	characteristic
558	C₂×C₄	#14524038	1	1	yes	characteristic
559	C₂×C₄	#14524038	1	1	yes	characteristic
560	C₂×C₄	#14524038	1	1	yes	characteristic
561	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
562	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
563	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
564	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
565	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
566	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
567	C₈	—	2	4	no	—
568	C₈	—	4	2	no	—
569	C₈	—	134	4	no	—
570	C₈	—	134	4	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	268	2	no	—
579	C₂²	—	536	2	no	—
580	C₂²	—	536	2	no	—
581	C₂²	—	536	2	no	—
582	C₂²	#16863285	1	1	yes	characteristic
583	C₂²	#16863285	1	1	yes	characteristic
584	C₂²	#16863285	1	1	yes	characteristic
585	C₂²	#16863285	1	1	yes	characteristic
586	C₂²	C₂×#14524000	1	1	yes	characteristic
587	C₂²	C₂×#14524028	1	1	yes	characteristic
588	C₂²	C₂×#14524029	1	1	yes	characteristic
589	C₄	—	8	2	no	—
590	C₄	—	268	4	no	—
591	C₄	—	268	2	no	—
592	C₄	#16863285	1	1	yes	characteristic
593	C₄	#16863285	1	1	yes	characteristic
594	C₄	#16863285	1	1	yes	characteristic
595	C₄	#16863285	1	1	yes	characteristic
596	C₄	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
597	C₄	C₂×#14524005	1	1	yes	characteristic
598	C₄	C₂×#14524016	1	1	yes	characteristic
599	C₄	C₂×#14524038	1	1	yes	characteristic
600	C₂	—	2	1	no	—
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	2	1	no	—
604	C₂	—	2	1	no	—
605	C₂	—	2	1	no	—
606	C₂	—	2	1	no	—
607	C₂	—	536	2	no	—
608	C₂	C₂×#16863285	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14524016
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#344×C₆₇	C₂
socle	C₂×C₆₇	C₂×#388
prime radical	C₂²×#11761628	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#344	D₆₇
hypocenter	C₆₇	#21080
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#344

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#16863444	C₂	semidirect
5. #22568627	C₂	semidirect
6. #22560876	C₂	semidirect
7. #22559918	C₂	semidirect
8. C₂×#344×C₆₇	C₂	semidirect
14. #16863444	C₂²	semidirect
22. #16863378	C₄	semidirect
24. C₂×#14523961	C₄	semidirect
25. C₂×#14523959	C₄	semidirect
28. M₁₆×Dic₆₇	C₂²	semidirect
29. #16863155	C₂²	semidirect
33. #344×C₆₇	C₂²	semidirect
74. #14523961	C₂×C₄	semidirect
75. #14523959	C₂×C₄	semidirect
311. C₂×#344	D₆₇	semidirect
313. C₆₇	#21080	semidirect
337. #344	C₂×D₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863444	2
C₂×#16863414	2
C₂×#16863412	2
C₂×#16863378	2
#22568627	2
#22560876	2
#22559918	2
C₂×#344×C₆₇	2
#21080	67