Subgroups — #30763365

Order 17152.

966 conjugacy classes of subgroups.

The subgroup lattice has 586 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×#16863402	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863400	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14523959	C₂	1	1	yes	maximal, characteristic
5	#22559947	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559919	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₂₆₈	C₂²	1	1	yes	characteristic
9	C₂²×#11761612	C₂²	1	1	yes	characteristic
10	C₂²×#11761612	C₂²	1	1	yes	characteristic
11	C₂×#14524002	C₂²	1	1	yes	characteristic
12	#16863444	—	2	2	no	—
13	#16863444	—	2	2	no	—
14	C₂×#14523978	—	2	2	no	—
15	#16863402	—	2	2	no	—
16	#16863402	—	2	2	no	—
17	#16863400	—	2	2	no	—
18	#16863400	—	2	2	no	—
19	C₂×#14523965	—	2	2	no	—
20	C₂×#14523963	—	2	2	no	—
21	C₂×#14523959	—	2	2	no	—
22	C₂×#14523959	—	2	2	no	—
23	C₂×#14523959	C₄	1	4	yes	complemented
24	#16863164	—	2	2	no	—
25	#16863156	—	2	2	no	—
26	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	#344×C₆₇	—	2	2	no	—
30	C₂⁴×D₆₇	—	2	1	no	—
31	C₂²×D₂₆₈	—	2	1	no	—
32	C₂²×D₂₆₈	—	2	1	no	—
33	C₂²×D₂₆₈	—	2	1	no	—
34	C₂²×D₂₆₈	—	2	1	no	—
35	C₂²×D₂₆₈	—	2	1	no	—
36	C₂²×D₂₆₈	—	2	1	no	—
37	C₂²×D₂₆₈	—	2	1	no	—
38	C₂²×D₂₆₈	—	2	1	no	—
39	C₂²×D₂₆₈	—	2	1	no	—
40	C₂²×D₂₆₈	—	2	1	no	—
41	C₂²×D₂₆₈	—	2	1	no	—
42	C₂²×D₂₆₈	C₂×C₄	1	2	yes	—
43	C₂²×D₂₆₈	D₄	1	2	yes	—
44	C₂²×D₂₆₈	Q₈	1	2	yes	—
45	C₂×#11761612	—	2	1	no	—
46	C₂×#11761612	—	2	1	no	—
47	C₂×#11761612	—	2	1	no	—
48	C₂×#11761612	—	2	1	no	—
49	C₂×#11761612	—	2	1	no	—
50	C₂×#11761612	—	2	1	no	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761612	—	2	1	no	—
53	C₂×#11761612	C₂×C₄	1	2	yes	—
54	C₂×#11761612	D₄	1	2	yes	—
55	C₂×#11761612	D₄	1	2	yes	—
56	C₂×#11761612	D₄	1	2	yes	—
57	C₂²×C₆₇⋊C₈	—	2	1	no	—
58	C₂²×C₆₇⋊C₈	—	2	1	no	—
59	#14524002	—	2	1	no	—
60	#14524002	—	2	1	no	—
61	#14524002	—	2	1	no	—
62	#14524002	—	2	1	no	—
63	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
64	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
65	C₂×C₄×Dic₆₇	—	2	1	no	—
66	#14523978	—	2	8	no	—
67	#14523965	—	2	8	no	—
68	#14523963	—	2	8	no	—
69	#14523959	—	2	8	no	—
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₆₇	—	2	1	no	—
72	C₂×M₁₆×C₆₇	—	2	1	no	—
73	C₂×M₁₆×C₆₇	—	2	1	no	—
74	C₂×M₁₆×C₆₇	—	2	1	no	—
75	C₂×M₁₆×C₆₇	—	2	1	no	—
76	C₂×M₁₆×C₆₇	—	2	1	no	—
77	C₂×M₁₆×C₆₇	—	2	1	no	—
78	C₂×M₁₆×C₆₇	—	2	1	no	—
79	C₂×M₁₆×C₆₇	—	2	1	no	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
82	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
83	C₂²×C₈×C₆₇	—	2	1	no	—
84	C₂²×C₈×C₆₇	—	2	1	no	—
85	#117×C₆₇	—	2	1	no	—
86	#117×C₆₇	—	2	1	no	—
87	#117×C₆₇	—	2	1	no	—
88	#117×C₆₇	D₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂³×D₆₇	—	2	2	no	—
93	C₂³×D₆₇	—	2	2	no	—
94	C₂³×D₆₇	—	2	2	no	—
95	C₂³×D₆₇	—	4	1	no	—
96	C₂³×D₆₇	—	4	1	no	—
97	C₂³×D₆₇	—	4	1	no	—
98	C₂³×D₆₇	—	4	1	no	—
99	C₂³×D₆₇	—	8	4	no	—
100	C₂×D₂₆₈	—	2	2	no	—
101	C₂×D₂₆₈	—	2	2	no	—
102	C₂×D₂₆₈	—	2	2	no	—
103	C₂×D₂₆₈	—	2	2	no	—
104	C₂×D₂₆₈	—	2	2	no	—
105	C₂×D₂₆₈	—	2	2	no	—
106	C₂×D₂₆₈	—	2	2	no	—
107	C₂×D₂₆₈	—	2	2	no	—
108	C₂×D₂₆₈	—	2	2	no	—
109	C₂×D₂₆₈	—	2	2	no	—
110	C₂×D₂₆₈	—	2	2	no	—
111	C₂×D₂₆₈	—	2	2	no	—
112	C₂×D₂₆₈	—	2	2	no	—
113	C₂×D₂₆₈	—	2	2	no	—
114	C₂×D₂₆₈	—	2	2	no	—
115	C₂×D₂₆₈	—	4	1	no	—
116	C₂×D₂₆₈	—	4	1	no	—
117	C₂×D₂₆₈	—	4	1	no	—
118	C₂×D₂₆₈	—	4	1	no	—
119	C₂×D₂₆₈	—	4	1	no	—
120	C₂×D₂₆₈	—	4	1	no	—
121	C₂×D₂₆₈	—	4	1	no	—
122	C₂×D₂₆₈	—	4	1	no	—
123	C₂×D₂₆₈	—	4	1	no	—
124	C₂×D₂₆₈	—	4	1	no	—
125	C₂×D₂₆₈	—	4	1	no	—
126	C₂×D₂₆₈	—	4	1	no	—
127	C₂×D₂₆₈	C₄⋊C₄	1	4	yes	—
128	C₂²×Dic₆₇	—	2	2	no	—
129	#11761612	—	2	2	no	—
130	#11761612	—	2	2	no	—
131	#11761612	—	2	2	no	—
132	#11761612	—	2	2	no	—
133	#11761612	—	2	2	no	—
134	#11761612	—	4	1	no	—
135	#11761612	—	4	1	no	—
136	#11761612	C₂²⋊C₄	1	4	yes	—
137	C₂×C₆₇⋊C₈	—	2	2	no	—
138	C₂×C₆₇⋊C₈	—	2	2	no	—
139	C₂×C₆₇⋊C₈	—	2	2	no	—
140	C₂×C₆₇⋊C₈	—	2	2	no	—
141	C₂×C₆₇⋊C₈	—	4	1	no	—
142	C₂×C₆₇⋊C₈	—	4	1	no	—
143	C₄⋊Dic₆₇	—	2	2	no	—
144	C₄⋊Dic₆₇	—	2	2	no	—
145	C₂²⋊Dic₆₇	—	4	1	no	—
146	C₂²⋊Dic₆₇	—	4	1	no	—
147	C₄×Dic₆₇	—	4	1	no	—
148	C₄×Dic₆₇	—	4	1	no	—
149	C₂⁴×C₆₇	#41	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	—	4	1	no	—
151	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
165	M₁₆×C₆₇	—	2	2	no	—
166	M₁₆×C₆₇	—	2	2	no	—
167	M₁₆×C₆₇	—	2	2	no	—
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	4	1	no	—
170	M₁₆×C₆₇	—	4	1	no	—
171	M₁₆×C₆₇	—	4	1	no	—
172	M₁₆×C₆₇	—	4	1	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	4	1	no	—
176	C₂×C₈×C₆₇	—	4	1	no	—
177	C₂×C₈×C₆₇	—	4	1	no	—
178	C₂×C₈×C₆₇	—	4	1	no	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	4	1	no	—
181	C₂²⋊C₄×C₆₇	—	2	2	no	—
182	C₂²⋊C₄×C₆₇	—	4	1	no	—
183	C₄²×C₆₇	—	2	2	no	—
184	C₄²×C₆₇	—	4	1	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	#117	1	1	yes	characteristic
187	C₂³×C₆₇	#121	1	1	yes	characteristic
188	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
189	C₂³×C₆₇	#123	1	1	yes	characteristic
190	C₂³×C₆₇	#123	1	1	yes	characteristic
191	C₂³×C₆₇	#124	1	1	yes	characteristic
192	C₂³×C₆₇	#124	1	1	yes	characteristic
193	C₂²×D₆₇	—	2	4	no	—
194	C₂²×D₆₇	—	4	2	no	—
195	C₂²×D₆₇	—	4	2	no	—
196	C₂²×D₆₇	—	4	2	no	—
197	C₂²×D₆₇	—	4	2	no	—
198	C₂²×D₆₇	—	4	2	no	—
199	C₂²×D₆₇	—	4	2	no	—
200	C₂²×D₆₇	—	8	4	no	—
201	C₂²×D₆₇	—	8	4	no	—
202	C₂²×D₆₇	—	8	4	no	—
203	C₂²×D₆₇	—	8	4	no	—
204	C₂²×D₆₇	—	8	4	no	—
205	C₂²×D₆₇	—	8	4	no	—
206	C₂²×D₆₇	—	8	4	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	8	2	no	—
209	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
216	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
217	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
218	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
219	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
220	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
221	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
222	C₂×C₄×C₆₇	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
231	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
232	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
233	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
234	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
235	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
236	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
237	C₂×Dic₆₇	—	4	2	no	—
238	C₂×Dic₆₇	—	8	2	no	—
239	D₂₆₈	—	2	4	no	—
240	D₂₆₈	—	2	4	no	—
241	D₂₆₈	—	4	2	no	—
242	D₂₆₈	—	4	2	no	—
243	D₂₆₈	—	4	2	no	—
244	D₂₆₈	—	4	2	no	—
245	D₂₆₈	—	4	2	no	—
246	D₂₆₈	—	4	2	no	—
247	C₈×C₆₇	—	4	2	no	—
248	C₈×C₆₇	—	4	2	no	—
249	C₆₇⋊C₈	—	2	4	no	—
250	C₆₇⋊C₈	—	4	2	no	—
251	C₂²×C₆₇	—	2	2	no	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	—	4	1	no	—
255	C₂²×C₆₇	—	4	1	no	—
256	C₂²×C₆₇	—	4	1	no	—
257	C₂²×C₆₇	—	4	1	no	—
258	C₂²×C₆₇	#380	1	1	yes	characteristic
259	C₂²×C₆₇	#388	1	1	yes	characteristic
260	C₂²×C₆₇	#388	1	1	yes	characteristic
261	C₂²×C₆₇	#393	1	1	yes	characteristic
262	C₂²×C₆₇	#394	1	1	yes	characteristic
263	C₂²×C₆₇	#394	1	1	yes	characteristic
264	C₂²×C₆₇	#399	1	1	yes	characteristic
265	C₂×D₆₇	—	4	4	no	—
266	C₂×D₆₇	—	8	4	no	—
267	C₂×D₆₇	—	8	4	no	—
268	C₂×D₆₇	—	8	4	no	—
269	C₂×D₆₇	—	8	4	no	—
270	C₂×D₆₇	—	8	4	no	—
271	C₂×D₆₇	—	8	4	no	—
272	C₂×D₆₇	—	8	4	no	—
273	C₄×C₆₇	—	8	2	no	—
274	C₄×C₆₇	#385	1	1	yes	characteristic
275	C₄×C₆₇	#385	1	1	yes	characteristic
276	C₄×C₆₇	#386	1	1	yes	characteristic
277	C₄×C₆₇	#390	1	1	yes	characteristic
278	C₄×C₆₇	#392	1	1	yes	characteristic
279	C₄×C₆₇	#396	1	1	yes	characteristic
280	C₄×C₆₇	#396	1	1	yes	characteristic
281	C₄×C₆₇	#397	1	1	yes	characteristic
282	Dic₆₇	—	8	2	no	—
283	#11142	—	67	1	no	maximal, Sylow
284	C₂×C₆₇	—	2	1	no	—
285	C₂×C₆₇	—	2	1	no	—
286	C₂×C₆₇	—	2	1	no	—
287	C₂×C₆₇	—	2	1	no	—
288	C₂×C₆₇	—	2	1	no	—
289	C₂×C₆₇	—	2	1	no	—
290	C₂×C₆₇	#1439	1	1	yes	characteristic
291	C₂×C₆₇	#2048	1	2	yes	—
292	D₆₇	—	8	4	no	—
293	C₂×#430	—	67	1	no	—
294	C₂×#418	—	67	1	no	—
295	C₂×#418	—	67	1	no	—
296	C₂²×#100	—	67	1	no	—
297	C₂×#344	D₆₇	1	1	yes	characteristic, complemented
298	#1742	—	67	1	no	—
299	#1733	—	67	1	no	—
300	C₆₇	#11142	1	1	yes	characteristic, complemented, Sylow
301	C₂³×D₄	—	67	1	no	—
302	C₂²×M₁₆	—	67	1	no	—
303	C₂²×M₁₆	—	67	1	no	—
304	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
305	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
306	C₂×#117	—	67	1	no	—
307	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
308	#430	—	134	2	no	—
309	#430	—	134	2	no	—
310	C₂×#108	—	134	2	no	—
311	#418	—	134	2	no	—
312	#418	—	134	2	no	—
313	#418	—	134	2	no	—
314	#418	—	134	2	no	—
315	C₂×#102	—	134	2	no	—
316	C₂×#102	—	134	2	no	—
317	C₂×#100	—	67	4	no	—
318	C₂×#100	—	134	2	no	—
319	C₂×#100	—	134	2	no	—
320	#344	—	2	2	no	—
321	#343	—	134	2	no	—
322	#339	—	134	2	no	—
323	C₂⁵	—	134	1	no	—
324	C₂²×D₄	—	67	2	no	—
325	C₂²×D₄	—	67	2	no	—
326	C₂²×D₄	—	67	2	no	—
327	C₂²×D₄	—	134	1	no	—
328	C₂²×D₄	—	134	1	no	—
329	C₂²×D₄	—	134	1	no	—
330	C₂²×D₄	—	134	1	no	—
331	C₂²×D₄	—	134	1	no	—
332	C₂²×D₄	—	134	1	no	—
333	C₂²×D₄	—	134	1	no	—
334	C₂²×D₄	—	134	1	no	—
335	C₂²×D₄	—	134	1	no	—
336	C₂²×D₄	—	134	1	no	—
337	C₂²×D₄	—	134	1	no	—
338	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
339	C₂×M₁₆	—	2	1	no	—
340	C₂×M₁₆	—	2	1	no	—
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	1	no	—
347	C₂×M₁₆	—	2	1	no	—
348	C₂×M₁₆	—	2	1	no	—
349	C₂×M₁₆	—	67	2	no	—
350	C₂×M₁₆	—	67	2	no	—
351	C₂×M₁₆	—	67	2	no	—
352	C₂×M₁₆	—	67	2	no	—
353	C₂×M₁₆	—	134	1	no	—
354	C₂×M₁₆	—	134	1	no	—
355	C₂×M₁₆	—	134	1	no	—
356	C₂×M₁₆	—	134	1	no	—
357	C₂×M₁₆	—	134	1	no	—
358	C₂×M₁₆	—	134	1	no	—
359	C₂×M₁₆	—	134	1	no	—
360	C₂×M₁₆	—	134	1	no	—
361	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
362	C₂×M₁₆	D₂₆₈	1	2	yes	—
363	C₂²×C₈	—	2	1	no	—
364	C₂²×C₈	—	2	1	no	—
365	C₂²×C₈	—	134	1	no	—
366	C₂²×C₈	—	134	1	no	—
367	#117	—	2	1	no	—
368	#117	—	2	1	no	—
369	#117	—	2	1	no	—
370	#117	—	134	1	no	—
371	#117	—	134	1	no	—
372	#117	—	134	1	no	—
373	#117	—	134	1	no	—
374	#117	C₂²⋊D₆₇	1	2	yes	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	67	2	no	—
377	C₂×C₂²⋊C₄	—	2	1	no	—
378	C₂×C₂²⋊C₄	—	134	1	no	—
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	—	134	1	no	—
381	#108	—	134	8	no	—
382	#102	—	134	8	no	—
383	#102	—	134	8	no	—
384	#100	—	134	8	no	—
385	C₂⁴	—	268	1	no	—
386	C₂⁴	—	268	1	no	—
387	C₂⁴	—	268	1	no	—
388	C₂⁴	—	268	1	no	—
389	C₂⁴	—	536	4	no	—
390	C₂⁴	—	134	2	no	—
391	C₂⁴	—	134	2	no	—
392	C₂⁴	—	134	2	no	—
393	C₂⁴	#11761628	1	1	yes	characteristic
394	C₂×D₄	—	268	1	no	—
395	C₂×D₄	—	268	1	no	—
396	C₂×D₄	—	268	1	no	—
397	C₂×D₄	—	268	1	no	—
398	C₂×D₄	—	268	1	no	—
399	C₂×D₄	—	268	1	no	—
400	C₂×D₄	—	268	1	no	—
401	C₂×D₄	—	268	1	no	—
402	C₂×D₄	—	268	1	no	—
403	C₂×D₄	—	268	1	no	—
404	C₂×D₄	—	268	1	no	—
405	C₂×D₄	—	268	1	no	—
406	C₂×D₄	—	67	4	no	—
407	C₂×D₄	—	134	2	no	—
408	C₂×D₄	—	134	2	no	—
409	C₂×D₄	—	134	2	no	—
410	C₂×D₄	—	134	2	no	—
411	C₂×D₄	—	134	2	no	—
412	C₂×D₄	—	134	2	no	—
413	C₂×D₄	—	134	2	no	—
414	C₂×D₄	—	134	2	no	—
415	C₂×D₄	—	134	2	no	—
416	C₂×D₄	—	134	2	no	—
417	C₂×D₄	—	134	2	no	—
418	C₂×D₄	—	134	2	no	—
419	C₂×D₄	—	134	2	no	—
420	C₂×D₄	—	134	2	no	—
421	C₂×D₄	—	134	2	no	—
422	C₂²×C₄	—	4	1	no	—
423	C₂²×C₄	—	134	2	no	—
424	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
425	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
426	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
427	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
428	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
429	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
432	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
433	C₂²×C₄	#11761627	1	1	yes	characteristic
434	C₂²×C₄	#11761628	1	1	yes	characteristic
435	C₂²×C₄	#11761628	1	1	yes	characteristic
436	C₂²×C₄	#11761628	1	1	yes	characteristic
437	C₂²×C₄	#11761629	1	1	yes	characteristic
438	M₁₆	—	2	2	no	—
439	M₁₆	—	2	2	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	4	1	no	—
443	M₁₆	—	4	1	no	—
444	M₁₆	—	4	1	no	—
445	M₁₆	—	4	1	no	—
446	M₁₆	—	268	1	no	—
447	M₁₆	—	268	1	no	—
448	M₁₆	—	67	4	no	—
449	M₁₆	—	134	2	no	—
450	M₁₆	—	134	2	no	—
451	M₁₆	—	134	2	no	—
452	M₁₆	—	134	2	no	—
453	M₁₆	—	134	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	4	1	no	—
457	C₂×C₈	—	4	1	no	—
458	C₂×C₈	—	4	1	no	—
459	C₂×C₈	—	4	1	no	—
460	C₂×C₈	—	268	1	no	—
461	C₂×C₈	—	268	1	no	—
462	C₂×C₈	—	134	2	no	—
463	C₂×C₈	—	134	2	no	—
464	C₂×C₈	—	134	2	no	—
465	C₂×C₈	—	134	2	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	4	1	no	—
468	C₄⋊C₄	—	134	2	no	—
469	C₄⋊C₄	—	134	2	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	4	1	no	—
472	C₂²⋊C₄	—	268	1	no	—
473	C₂²⋊C₄	—	268	1	no	—
474	C₄²	—	2	2	no	—
475	C₄²	—	4	1	no	—
476	C₄²	—	268	1	no	—
477	C₄²	—	268	1	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	268	2	no	—
480	C₂³	—	268	2	no	—
481	C₂³	—	268	2	no	—
482	C₂³	—	268	2	no	—
483	C₂³	—	268	2	no	—
484	C₂³	—	268	2	no	—
485	C₂³	—	536	4	no	—
486	C₂³	—	536	4	no	—
487	C₂³	—	536	4	no	—
488	C₂³	—	536	4	no	—
489	C₂³	—	536	4	no	—
490	C₂³	—	536	4	no	—
491	C₂³	—	536	4	no	—
492	C₂³	—	134	4	no	—
493	C₂³	#14524000	1	1	yes	characteristic
494	C₂³	#14524016	1	1	yes	characteristic
495	C₂³	#14524021	1	1	yes	characteristic
496	C₂³	#14524026	1	1	yes	characteristic
497	C₂³	#14524028	1	1	yes	characteristic
498	C₂³	#14524029	1	1	yes	characteristic
499	C₂³	#14524031	1	1	yes	characteristic
500	D₄	—	268	2	no	—
501	D₄	—	268	2	no	—
502	D₄	—	268	2	no	—
503	D₄	—	268	2	no	—
504	D₄	—	268	2	no	—
505	D₄	—	268	2	no	—
506	D₄	—	134	4	no	—
507	D₄	—	134	4	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	8	2	no	—
510	C₂×C₄	—	268	2	no	—
511	C₂×C₄	—	536	2	no	—
512	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
513	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
514	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
515	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
516	C₂×C₄	#14524005	1	1	yes	characteristic
517	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
518	C₂×C₄	#14524011	1	1	yes	characteristic
519	C₂×C₄	#14524013	1	1	yes	characteristic
520	C₂×C₄	#14524013	1	1	yes	characteristic
521	C₂×C₄	#14524015	1	1	yes	characteristic
522	C₂×C₄	#14524016	1	1	yes	characteristic
523	C₂×C₄	#14524016	1	1	yes	characteristic
524	C₂×C₄	#14524016	1	1	yes	characteristic
525	C₂×C₄	#14524016	1	1	yes	characteristic
526	C₂×C₄	#14524017	1	1	yes	characteristic
527	C₂×C₄	#14524018	1	1	yes	characteristic
528	C₂×C₄	#14524019	1	1	yes	characteristic
529	C₂×C₄	#14524021	1	1	yes	characteristic
530	C₂×C₄	#14524021	1	1	yes	characteristic
531	C₂×C₄	#14524021	1	1	yes	characteristic
532	C₂×C₄	#14524023	1	1	yes	characteristic
533	C₂×C₄	#14524025	1	1	yes	characteristic
534	C₂×C₄	#14524028	1	1	yes	characteristic
535	C₂×C₄	#14524028	1	1	yes	characteristic
536	C₂×C₄	#14524029	1	1	yes	characteristic
537	C₂×C₄	#14524037	1	1	yes	characteristic
538	C₂×C₄	#14524038	1	1	yes	characteristic
539	C₂×C₄	#14524039	1	1	yes	characteristic
540	C₈	—	4	2	no	—
541	C₈	—	4	2	no	—
542	C₈	—	268	2	no	—
543	C₈	—	134	4	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	268	4	no	—
552	C₂²	—	536	4	no	—
553	C₂²	—	536	4	no	—
554	C₂²	—	536	4	no	—
555	C₂²	—	536	4	no	—
556	C₂²	—	536	4	no	—
557	C₂²	—	536	4	no	—
558	C₂²	—	536	4	no	—
559	C₂²	#16863255	1	1	yes	characteristic
560	C₂²	#16863285	1	1	yes	characteristic
561	C₂²	#16863287	1	1	yes	characteristic
562	C₂²	#16863309	1	1	yes	characteristic
563	C₂²	#16863311	1	1	yes	characteristic
564	C₂²	#16863314	1	1	yes	characteristic
565	C₂²	#16863331	1	1	yes	characteristic
566	C₄	—	8	2	no	—
567	C₄	—	536	2	no	—
568	C₄	#16863267	1	1	yes	characteristic
569	C₄	#16863270	1	1	yes	characteristic
570	C₄	#16863274	1	1	yes	characteristic
571	C₄	#16863296	1	1	yes	characteristic
572	C₄	#16863305	1	1	yes	characteristic
573	C₄	#16863318	1	1	yes	characteristic
574	C₄	#16863320	1	1	yes	characteristic
575	C₄	#16863325	1	1	yes	characteristic
576	C₂	—	2	1	no	—
577	C₂	—	2	1	no	—
578	C₂	—	2	1	no	—
579	C₂	—	2	1	no	—
580	C₂	—	2	1	no	—
581	C₂	—	2	1	no	—
582	C₂	—	536	4	no	—
583	C₂	#22558453	1	1	yes	characteristic
584	C₂	#22561485	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863314
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#344×C₆₇	C₂
socle	C₂²×C₆₇	#394
prime radical	C₂³×D₂₆₈	C₂²
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#344	D₆₇
hypocenter	C₆₇	#11142
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ 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. #22559947	C₂	semidirect
6. #22559919	C₂	semidirect
7. C₂×#344×C₆₇	C₂	semidirect
23. C₂×#14523959	C₄	semidirect
297. C₂×#344	D₆₇	semidirect
300. C₆₇	#11142	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×#16863402	2
C₂×#16863400	2
C₂²×#14523959	2
#22559947	2
#22559919	2
C₂×#344×C₆₇	2
#11142	67