Subgroups — #30763137

Order 17152.

866 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523978	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×#16863364	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863168	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559925	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761628	C₂²	1	1	yes	characteristic
9	C₂²×#11761612	C₂²	1	1	yes	characteristic
10	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#14524002	C₂²	1	1	yes	characteristic
12	C₂×#14523978	—	2	2	no	—
13	C₂×#14523978	—	2	2	no	—
14	C₂×#14523978	—	2	2	no	—
15	#16863414	—	2	2	no	—
16	#16863414	—	2	2	no	—
17	#16863412	—	2	2	no	—
18	#16863412	—	2	2	no	—
19	C₂×#14523971	—	2	2	no	—
20	C₂×#14523969	—	2	2	no	—
21	#16863364	—	2	2	no	—
22	#16863364	C₄	1	4	yes	complemented
23	C₂×(C₂×D₆₇)⋊C₈	—	2	2	no	—
24	#16863168	—	2	2	no	—
25	#16863158	—	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	C₂×#11761628	—	2	1	no	—
31	C₂×#11761628	—	2	1	no	—
32	C₂×#11761628	—	2	1	no	—
33	C₂×#11761628	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
41	C₂×#11761628	D₄	1	2	yes	—
42	C₂²×Dic₁₃₄	Q₈	1	1	yes	characteristic
43	C₂²×D₂₆₈	Q₈	1	1	yes	characteristic
44	C₂²×C₂²⋊D₆₇	—	2	1	no	—
45	C₂²×C₄×D₆₇	—	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	D₄	1	2	yes	—
52	C₂²×C₆₇⋊C₈	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	—	2	1	no	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	—	2	1	no	—
56	C₂²×C₆₇⋊C₈	—	2	1	no	—
57	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
58	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
59	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
60	#14524002	—	2	1	no	—
61	#14524002	—	2	1	no	—
62	#14524002	—	2	1	no	—
63	#14524002	D₄	1	2	yes	—
64	C₂×C₄⋊Dic₆₇	—	2	1	no	—
65	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
66	C₂×C₄×Dic₆₇	—	2	1	no	—
67	#14523978	—	2	8	no	—
68	#14523971	—	2	8	no	—
69	#14523969	—	2	8	no	—
70	(C₂×D₆₇)⋊C₈	—	2	8	no	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
82	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
84	C₂²×C₈×C₆₇	—	2	1	no	—
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	#117×C₆₇	—	2	1	no	—
87	#117×C₆₇	—	2	1	no	—
88	#117×C₆₇	—	2	1	no	—
89	#117×C₆₇	D₄	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	—	2	1	no	—
93	C₂³×D₆₇	—	2	1	no	—
94	#11761628	—	2	2	no	—
95	#11761628	—	2	2	no	—
96	#11761628	—	2	2	no	—
97	#11761628	—	2	2	no	—
98	#11761628	—	2	2	no	—
99	#11761628	—	2	2	no	—
100	#11761628	—	2	2	no	—
101	#11761628	—	2	2	no	—
102	#11761628	—	4	1	no	—
103	#11761628	—	4	1	no	—
104	#11761628	—	4	1	no	—
105	#11761628	—	4	1	no	—
106	#11761628	—	4	1	no	—
107	#11761628	—	4	1	no	—
108	#11761628	—	4	1	no	—
109	#11761628	—	4	1	no	—
110	C₂×Dic₁₃₄	—	2	1	no	—
111	C₂×Dic₁₃₄	—	2	1	no	—
112	C₂×Dic₁₃₄	—	2	1	no	—
113	C₂×Dic₁₃₄	—	2	1	no	—
114	C₂×Dic₁₃₄	—	2	1	no	—
115	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
116	C₂×D₂₆₈	—	2	1	no	—
117	C₂×D₂₆₈	—	2	1	no	—
118	C₂×D₂₆₈	—	2	1	no	—
119	C₂×D₂₆₈	—	2	1	no	—
120	C₂×D₂₆₈	—	2	1	no	—
121	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
122	C₂×C₂²⋊D₆₇	—	2	2	no	—
123	C₂×C₂²⋊D₆₇	—	2	2	no	—
124	C₂×C₂²⋊D₆₇	—	4	1	no	—
125	C₂×C₂²⋊D₆₇	—	4	1	no	—
126	C₂×C₂²⋊D₆₇	—	4	1	no	—
127	C₂×C₂²⋊D₆₇	—	4	1	no	—
128	C₂²×Dic₆₇	—	2	1	no	—
129	C₂²×Dic₆₇	—	4	1	no	—
130	C₂×C₄×D₆₇	—	2	2	no	—
131	C₂×C₄×D₆₇	—	2	2	no	—
132	C₂×C₄×D₆₇	—	4	1	no	—
133	C₂×C₄×D₆₇	—	4	1	no	—
134	C₂×C₄×D₆₇	—	4	1	no	—
135	C₂×C₄×D₆₇	—	4	1	no	—
136	#11761612	—	2	2	no	—
137	#11761612	—	2	2	no	—
138	#11761612	—	4	1	no	—
139	#11761612	—	4	1	no	—
140	C₂×C₆₇⋊C₈	—	2	2	no	—
141	C₂×C₆₇⋊C₈	—	2	2	no	—
142	C₂×C₆₇⋊C₈	—	2	2	no	—
143	C₂×C₆₇⋊C₈	—	2	2	no	—
144	C₂×C₆₇⋊C₈	—	2	2	no	—
145	C₂×C₆₇⋊C₈	—	2	2	no	—
146	C₂×C₆₇⋊C₈	—	2	2	no	—
147	C₂×C₆₇⋊C₈	—	4	1	no	—
148	C₂×C₆₇⋊C₈	—	4	1	no	—
149	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	4	yes	—
150	C₄⋊Dic₆₇	—	2	2	no	—
151	C₄⋊Dic₆₇	—	4	1	no	—
152	C₂²⋊Dic₆₇	—	2	2	no	—
153	C₂²⋊Dic₆₇	—	4	1	no	—
154	C₄×Dic₆₇	—	2	2	no	—
155	C₄×Dic₆₇	—	4	1	no	—
156	C₂⁴×C₆₇	#41	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	—	4	1	no	—
158	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
172	M₁₆×C₆₇	—	2	2	no	—
173	M₁₆×C₆₇	—	2	2	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	—	4	1	no	—
177	M₁₆×C₆₇	—	4	1	no	—
178	M₁₆×C₆₇	—	4	1	no	—
179	M₁₆×C₆₇	—	4	1	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	—	2	2	no	—
182	C₂×C₈×C₆₇	—	4	1	no	—
183	C₂×C₈×C₆₇	—	4	1	no	—
184	C₂×C₈×C₆₇	—	4	1	no	—
185	C₂×C₈×C₆₇	—	4	1	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	4	1	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	4	1	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	4	1	no	—
192	C₂³×C₆₇	—	4	2	no	—
193	C₂³×C₆₇	#117	1	1	yes	characteristic
194	C₂³×C₆₇	#121	1	1	yes	characteristic
195	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
196	C₂³×C₆₇	#123	1	1	yes	characteristic
197	C₂³×C₆₇	#123	1	1	yes	characteristic
198	C₂³×C₆₇	#124	1	1	yes	characteristic
199	C₂³×C₆₇	#124	1	1	yes	characteristic
200	C₂²×D₆₇	—	2	2	no	—
201	C₂²×D₆₇	—	4	4	no	—
202	C₂²×D₆₇	—	4	1	no	—
203	C₂²×D₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	8	2	no	—
206	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
213	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
214	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
215	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
216	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
217	C₂×C₄×C₆₇	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²⋊Q₈	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	#123	1	1	yes	characteristic
228	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
229	C₂×C₄×C₆₇	#123	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	#127	1	1	yes	characteristic
234	C₂²⋊D₆₇	—	4	2	no	—
235	C₂²⋊D₆₇	—	4	2	no	—
236	C₂²⋊D₆₇	—	4	2	no	—
237	C₂²⋊D₆₇	—	4	2	no	—
238	C₂×Dic₆₇	—	2	2	no	—
239	C₂×Dic₆₇	—	4	1	no	—
240	C₂×Dic₆₇	—	4	1	no	—
241	C₂×Dic₆₇	—	4	2	no	—
242	C₂×Dic₆₇	—	8	2	no	—
243	D₂₆₈	—	2	2	no	—
244	D₂₆₈	—	2	2	no	—
245	D₂₆₈	—	4	1	no	—
246	D₂₆₈	—	4	1	no	—
247	C₄×D₆₇	—	4	2	no	—
248	C₄×D₆₇	—	4	2	no	—
249	C₄×D₆₇	—	4	2	no	—
250	C₄×D₆₇	—	4	2	no	—
251	Dic₁₃₄	—	2	2	no	—
252	Dic₁₃₄	—	2	2	no	—
253	Dic₁₃₄	—	4	1	no	—
254	Dic₁₃₄	—	4	1	no	—
255	C₈×C₆₇	—	4	2	no	—
256	C₈×C₆₇	—	4	2	no	—
257	C₆₇⋊C₈	—	2	4	no	—
258	C₆₇⋊C₈	—	4	2	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	4	1	no	—
263	C₂²×C₆₇	—	4	1	no	—
264	C₂²×C₆₇	—	4	1	no	—
265	C₂²×C₆₇	—	4	1	no	—
266	C₂²×C₆₇	#380	1	1	yes	characteristic
267	C₂²×C₆₇	#388	1	1	yes	characteristic
268	C₂²×C₆₇	#388	1	1	yes	characteristic
269	C₂²×C₆₇	#393	1	1	yes	characteristic
270	C₂²×C₆₇	#394	1	1	yes	characteristic
271	C₂²×C₆₇	#394	1	1	yes	characteristic
272	C₂²×C₆₇	#399	1	1	yes	characteristic
273	C₂×D₆₇	—	4	4	no	—
274	C₂×D₆₇	—	4	2	no	—
275	C₂×D₆₇	—	8	2	no	—
276	C₂×D₆₇	—	8	2	no	—
277	C₄×C₆₇	—	8	2	no	—
278	C₄×C₆₇	#385	1	1	yes	characteristic
279	C₄×C₆₇	#385	1	1	yes	characteristic
280	C₄×C₆₇	#386	1	1	yes	characteristic
281	C₄×C₆₇	#390	1	1	yes	characteristic
282	C₄×C₆₇	#392	1	1	yes	characteristic
283	C₄×C₆₇	#396	1	1	yes	characteristic
284	C₄×C₆₇	#396	1	1	yes	characteristic
285	C₄×C₆₇	#397	1	1	yes	characteristic
286	Dic₆₇	—	4	2	no	—
287	Dic₆₇	—	8	2	no	—
288	#11108	—	67	1	no	maximal, Sylow
289	C₂×C₆₇	—	2	1	no	—
290	C₂×C₆₇	—	2	1	no	—
291	C₂×C₆₇	—	2	1	no	—
292	C₂×C₆₇	—	2	1	no	—
293	C₂×C₆₇	—	2	1	no	—
294	C₂×C₆₇	—	2	1	no	—
295	C₂×C₆₇	#1439	1	1	yes	characteristic
296	C₂×C₆₇	#1814	1	2	yes	—
297	D₆₇	—	8	2	no	—
298	C₂²×#108	—	67	1	no	—
299	C₂×#421	—	67	1	no	—
300	C₂×#421	—	67	1	no	—
301	C₂×#408	—	67	1	no	—
302	C₂×#344	—	67	1	no	—
303	C₂×#344	D₆₇	1	1	yes	characteristic, complemented
304	#1735	—	67	1	no	—
305	C₆₇	#11108	1	1	yes	characteristic, complemented, Sylow
306	C₂²×#41	—	67	1	no	—
307	C₂²×M₁₆	—	67	1	no	—
308	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
309	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
310	C₂³×C₈	—	67	1	no	—
311	C₂×#117	—	67	1	no	—
312	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
313	C₂×#108	—	134	2	no	—
314	C₂×#108	—	134	2	no	—
315	C₂×#108	—	134	2	no	—
316	#421	—	134	2	no	—
317	#421	—	134	2	no	—
318	#421	—	134	2	no	—
319	#421	—	134	2	no	—
320	C₂×#104	—	134	2	no	—
321	C₂×#104	—	134	2	no	—
322	#408	—	67	4	no	—
323	#408	—	134	2	no	—
324	C₂×C₂²⋊C₈	—	134	2	no	—
325	#344	—	2	2	no	—
326	#344	—	134	2	no	—
327	#340	—	134	2	no	—
328	C₂×#41	—	67	2	no	—
329	C₂×#41	—	67	2	no	—
330	C₂×#41	—	134	1	no	—
331	C₂×#41	—	134	1	no	—
332	C₂×#41	—	134	1	no	—
333	C₂×#41	—	134	1	no	—
334	C₂×#41	—	134	1	no	—
335	C₂×#41	—	134	1	no	—
336	C₂×#41	—	134	1	no	—
337	C₂×#41	—	134	1	no	—
338	C₂×#41	—	134	1	no	—
339	C₂×#41	—	134	1	no	—
340	C₂²×Q₈	—	67	1	no	—
341	C₂²×D₄	—	67	1	no	—
342	C₂²×D₄	—	134	1	no	—
343	C₂³×C₄	—	134	1	no	—
344	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	67	2	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	—	67	2	no	—
366	C₂²×C₈	—	67	2	no	—
367	C₂²×C₈	—	67	2	no	—
368	C₂²×C₈	—	134	1	no	—
369	C₂²×C₈	—	134	1	no	—
370	C₂²×C₈	—	134	1	no	—
371	C₂²×C₈	—	134	1	no	—
372	C₂²×C₈	—	134	1	no	—
373	#117	—	2	1	no	—
374	#117	—	2	1	no	—
375	#117	—	2	1	no	—
376	#117	—	67	2	no	—
377	#117	—	134	1	no	—
378	#117	—	134	1	no	—
379	#117	—	134	1	no	—
380	#117	C₂²⋊D₆₇	1	2	yes	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	134	1	no	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	134	1	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	—	134	1	no	—
387	#108	—	134	8	no	—
388	#104	—	134	8	no	—
389	#104	—	134	8	no	—
390	C₂²⋊C₈	—	134	8	no	—
391	C₂⁴	—	134	1	no	—
392	C₂⁴	#11761628	1	1	yes	characteristic
393	#41	—	268	1	no	—
394	#41	—	268	1	no	—
395	#41	—	268	1	no	—
396	#41	—	268	1	no	—
397	#41	—	268	1	no	—
398	#41	—	268	1	no	—
399	#41	—	268	1	no	—
400	#41	—	268	1	no	—
401	#41	—	134	2	no	—
402	#41	—	134	2	no	—
403	#41	—	134	2	no	—
404	#41	—	134	2	no	—
405	#41	—	134	2	no	—
406	#41	—	134	2	no	—
407	#41	—	134	2	no	—
408	#41	—	134	2	no	—
409	C₂×Q₈	—	67	2	no	—
410	C₂×Q₈	—	134	1	no	—
411	C₂×Q₈	—	134	1	no	—
412	C₂×Q₈	—	134	1	no	—
413	C₂×Q₈	—	134	1	no	—
414	C₂×Q₈	—	134	1	no	—
415	C₂×D₄	—	268	1	no	—
416	C₂×D₄	—	268	1	no	—
417	C₂×D₄	—	268	1	no	—
418	C₂×D₄	—	268	1	no	—
419	C₂×D₄	—	67	2	no	—
420	C₂×D₄	—	134	2	no	—
421	C₂×D₄	—	134	2	no	—
422	C₂×D₄	—	134	1	no	—
423	C₂×D₄	—	134	1	no	—
424	C₂×D₄	—	134	1	no	—
425	C₂×D₄	—	134	1	no	—
426	C₂×D₄	—	134	1	no	—
427	C₂²×C₄	—	4	1	no	—
428	C₂²×C₄	—	268	1	no	—
429	C₂²×C₄	—	268	1	no	—
430	C₂²×C₄	—	268	1	no	—
431	C₂²×C₄	—	268	1	no	—
432	C₂²×C₄	—	268	1	no	—
433	C₂²×C₄	—	134	1	no	—
434	C₂²×C₄	—	134	2	no	—
435	C₂²×C₄	—	134	2	no	—
436	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
437	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
438	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
439	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
440	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
441	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
444	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
445	C₂²×C₄	#11761627	1	1	yes	characteristic
446	C₂²×C₄	#11761628	1	1	yes	characteristic
447	C₂²×C₄	#11761628	1	1	yes	characteristic
448	C₂²×C₄	#11761628	1	1	yes	characteristic
449	C₂²×C₄	#11761629	1	1	yes	characteristic
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	—	2	2	no	—
454	M₁₆	—	4	1	no	—
455	M₁₆	—	4	1	no	—
456	M₁₆	—	4	1	no	—
457	M₁₆	—	4	1	no	—
458	M₁₆	—	268	1	no	—
459	M₁₆	—	268	1	no	—
460	M₁₆	—	134	2	no	—
461	M₁₆	—	134	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	4	1	no	—
465	C₂×C₈	—	4	1	no	—
466	C₂×C₈	—	4	1	no	—
467	C₂×C₈	—	4	1	no	—
468	C₂×C₈	—	268	1	no	—
469	C₂×C₈	—	268	1	no	—
470	C₂×C₈	—	67	4	no	—
471	C₂×C₈	—	134	2	no	—
472	C₂×C₈	—	134	2	no	—
473	C₂×C₈	—	134	2	no	—
474	C₂×C₈	—	134	2	no	—
475	C₂×C₈	—	134	2	no	—
476	C₂×C₈	—	134	2	no	—
477	C₂×C₈	—	134	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	268	1	no	—
481	C₄⋊C₄	—	134	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	268	1	no	—
485	C₂²⋊C₄	—	134	2	no	—
486	C₄²	—	2	2	no	—
487	C₄²	—	4	1	no	—
488	C₄²	—	268	1	no	—
489	C₄²	—	134	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	268	4	no	—
492	C₂³	—	268	1	no	—
493	C₂³	—	268	1	no	—
494	C₂³	—	134	2	no	—
495	C₂³	#14524000	1	1	yes	characteristic
496	C₂³	#14524016	1	1	yes	characteristic
497	C₂³	#14524021	1	1	yes	characteristic
498	C₂³	#14524026	1	1	yes	characteristic
499	C₂³	#14524028	1	1	yes	characteristic
500	C₂³	#14524029	1	1	yes	characteristic
501	C₂³	#14524031	1	1	yes	characteristic
502	Q₈	—	268	1	no	—
503	Q₈	—	268	1	no	—
504	Q₈	—	134	2	no	—
505	Q₈	—	134	2	no	—
506	D₄	—	268	2	no	—
507	D₄	—	268	2	no	—
508	D₄	—	268	2	no	—
509	D₄	—	268	2	no	—
510	D₄	—	268	1	no	—
511	D₄	—	268	1	no	—
512	D₄	—	134	2	no	—
513	D₄	—	134	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	8	2	no	—
516	C₂×C₄	—	268	1	no	—
517	C₂×C₄	—	268	1	no	—
518	C₂×C₄	—	268	2	no	—
519	C₂×C₄	—	268	2	no	—
520	C₂×C₄	—	268	2	no	—
521	C₂×C₄	—	268	2	no	—
522	C₂×C₄	—	268	2	no	—
523	C₂×C₄	—	536	2	no	—
524	C₂×C₄	—	134	2	no	—
525	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
526	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
527	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
528	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
529	C₂×C₄	#14524005	1	1	yes	characteristic
530	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
531	C₂×C₄	#14524011	1	1	yes	characteristic
532	C₂×C₄	#14524013	1	1	yes	characteristic
533	C₂×C₄	#14524013	1	1	yes	characteristic
534	C₂×C₄	#14524015	1	1	yes	characteristic
535	C₂×C₄	#14524016	1	1	yes	characteristic
536	C₂×C₄	#14524016	1	1	yes	characteristic
537	C₂×C₄	#14524016	1	1	yes	characteristic
538	C₂×C₄	#14524016	1	1	yes	characteristic
539	C₂×C₄	#14524017	1	1	yes	characteristic
540	C₂×C₄	#14524018	1	1	yes	characteristic
541	C₂×C₄	#14524019	1	1	yes	characteristic
542	C₂×C₄	#14524021	1	1	yes	characteristic
543	C₂×C₄	#14524021	1	1	yes	characteristic
544	C₂×C₄	#14524021	1	1	yes	characteristic
545	C₂×C₄	#14524023	1	1	yes	characteristic
546	C₂×C₄	#14524025	1	1	yes	characteristic
547	C₂×C₄	#14524028	1	1	yes	characteristic
548	C₂×C₄	#14524028	1	1	yes	characteristic
549	C₂×C₄	#14524029	1	1	yes	characteristic
550	C₂×C₄	#14524037	1	1	yes	characteristic
551	C₂×C₄	#14524038	1	1	yes	characteristic
552	C₂×C₄	#14524039	1	1	yes	characteristic
553	C₈	—	4	2	no	—
554	C₈	—	4	2	no	—
555	C₈	—	268	2	no	—
556	C₈	—	134	4	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	4	1	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	268	4	no	—
565	C₂²	—	268	2	no	—
566	C₂²	—	536	2	no	—
567	C₂²	—	536	2	no	—
568	C₂²	#16863255	1	1	yes	characteristic
569	C₂²	#16863285	1	1	yes	characteristic
570	C₂²	#16863287	1	1	yes	characteristic
571	C₂²	#16863309	1	1	yes	characteristic
572	C₂²	#16863311	1	1	yes	characteristic
573	C₂²	#16863314	1	1	yes	characteristic
574	C₂²	#16863331	1	1	yes	characteristic
575	C₄	—	8	2	no	—
576	C₄	—	268	2	no	—
577	C₄	—	536	2	no	—
578	C₄	#16863267	1	1	yes	characteristic
579	C₄	#16863270	1	1	yes	characteristic
580	C₄	#16863274	1	1	yes	characteristic
581	C₄	#16863296	1	1	yes	characteristic
582	C₄	#16863305	1	1	yes	characteristic
583	C₄	#16863318	1	1	yes	characteristic
584	C₄	#16863320	1	1	yes	characteristic
585	C₄	#16863325	1	1	yes	characteristic
586	C₂	—	2	1	no	—
587	C₂	—	2	1	no	—
588	C₂	—	2	1	no	—
589	C₂	—	2	1	no	—
590	C₂	—	2	1	no	—
591	C₂	—	2	1	no	—
592	C₂	—	536	2	no	—
593	C₂	#22558453	1	1	yes	characteristic
594	C₂	#22560262	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524016
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#344×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂²×#11761628	C₂²
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#344	D₆₇
hypocenter	C₆₇	#11108
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#344

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523978	C₂	semidirect
5. C₂×#16863168	C₂	semidirect
6. #22559925	C₂	semidirect
7. C₂×#344×C₆₇	C₂	semidirect
22. #16863364	C₄	semidirect
303. C₂×#344	D₆₇	semidirect
305. C₆₇	#11108	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523978	2
C₂×#16863414	2
C₂×#16863412	2
C₂×#16863364	2
C₂×#16863168	2
#22559925	2
C₂×#344×C₆₇	2
#11108	67