Subgroups — #30825651

Order 17152.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569322	C₂	1	1	yes	maximal, characteristic
2	C₄×#14524074	C₂	1	1	yes	maximal, characteristic
3	#22559116	C₂	1	2	yes	maximal, complemented
4	#22559109	C₂	1	2	yes	maximal, complemented
5	#22559104	C₂	1	2	yes	maximal, complemented
6	#22559099	C₂	1	2	yes	maximal, complemented
7	#22558962	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558589	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2983	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524074	C₄	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863526	C₂²	1	2	yes	—
15	#16863501	C₄	1	2	yes	—
16	#16863494	C₄	1	2	yes	—
17	C₄×#11761618	C₂²	1	2	yes	—
18	C₄×#11761615	C₂²	1	2	yes	—
19	#16863475	C₂²	1	2	yes	—
20	#16863464	—	2	2	no	—
21	#16863452	—	2	1	no	—
22	#16863436	C₄	1	1	yes	characteristic
23	#16863418	—	2	1	no	—
24	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
25	#16863405	C₄	1	1	yes	characteristic
26	#16863395	C₄	1	1	yes	characteristic
27	C₄×#11761612	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
29	#16863148	C₂²	1	2	yes	—
30	#16863146	C₂²	1	2	yes	—
31	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
32	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
33	#16863126	C₂²	1	1	yes	characteristic
34	#16863126	C₂²	1	1	yes	characteristic
35	#16863123	C₂²	1	1	yes	characteristic
36	#16863123	C₂²	1	1	yes	characteristic
37	#16863118	C₂²	1	2	yes	—
38	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
39	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	—	2	1	no	—
41	#446×C₆₇	—	2	1	no	—
42	#446×C₆₇	C₂²	1	2	yes	—
43	#446×C₆₇	C₂²	1	2	yes	—
44	#432×C₆₇	—	2	2	no	—
45	#431×C₆₇	C₂²	1	1	yes	characteristic
46	#431×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
48	#14524074	—	2	4	no	—
49	C₂×#11761618	C₂×C₄	1	2	yes	—
50	C₂×#11761615	C₂×C₄	1	2	yes	—
51	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
52	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761611	—	2	2	no	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	—	2	1	no	—
56	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
58	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
59	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
60	C₄×C₂²⋊D₆₇	—	2	1	no	—
61	C₄×C₂²⋊D₆₇	—	2	1	no	—
62	#14524002	C₂×C₄	1	1	yes	characteristic
63	#14524000	—	2	1	no	—
64	C₄²×D₆₇	—	2	1	no	—
65	#14523976	C₂×C₄	1	2	yes	—
66	#14523974	C₂×C₄	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
69	Dic₆₇⋊C₈	—	2	2	no	—
70	#14523967	C₂×C₄	1	2	yes	—
71	#14523967	C₂×C₄	1	2	yes	—
72	#14523966	C₂×C₄	1	2	yes	—
73	#14523963	C₂×C₄	1	2	yes	—
74	(C₂×D₆₇)⋊C₈	—	2	2	no	—
75	#14523954	C₂×C₄	1	2	yes	—
76	#14523953	—	2	2	no	—
77	C₄×C₆₇⋊C₈	—	2	1	no	—
78	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
81	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₆₇	—	2	1	no	—
83	C₄×Q₈×C₆₇	—	2	1	no	—
84	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₆₇	—	2	2	no	—
91	C₄⋊C₈×C₆₇	—	2	2	no	—
92	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
98	#97×C₆₇	—	2	2	no	—
99	#97×C₆₇	—	2	2	no	—
100	C₄×C₈×C₆₇	—	2	1	no	—
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	#11761628	—	2	2	no	—
103	#11761628	—	2	2	no	—
104	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
105	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₂²⋊D₆₇	—	2	1	no	—
107	C₂×C₄×D₆₇	—	2	1	no	—
108	C₂×C₄×D₆₇	—	2	1	no	—
109	C₂×C₄×D₆₇	—	2	1	no	—
110	#11761618	—	2	4	no	—
111	#11761615	—	2	4	no	—
112	#11761612	—	2	2	no	—
113	#11761612	M₁₆	1	4	yes	complemented
114	#11761611	—	8	2	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	—	2	1	no	—
118	C₂×C₆₇⋊C₈	—	4	2	no	—
119	C₂×C₆₇⋊C₈	—	8	1	no	—
120	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
121	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
122	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
124	Dic₆₇⋊C₄	—	2	1	no	—
125	Dic₆₇⋊C₄	—	2	1	no	—
126	C₂²⋊Dic₆₇	—	2	1	no	—
127	(C₂×D₆₇)⋊C₄	—	2	1	no	—
128	(C₂×D₆₇)⋊C₄	—	2	1	no	—
129	C₄×Dic₆₇	—	2	1	no	—
130	C₄×Dic₆₇	—	2	1	no	—
131	C₄×Dic₆₇	—	2	1	no	—
132	C₂×Q₈×C₆₇	—	2	1	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₆₇	#41	1	1	yes	characteristic
154	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
155	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
156	C₂²×D₆₇	—	2	1	no	—
157	Q₈×C₆₇	—	2	2	no	—
158	Q₈×C₆₇	—	2	2	no	—
159	Q₈×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	8	1	no	—
167	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
171	C₂×C₄×C₆₇	#131	1	2	yes	—
172	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
175	C₂²⋊D₆₇	—	4	2	no	—
176	C₂×Dic₆₇	—	2	1	no	—
177	C₂×Dic₆₇	—	2	1	no	—
178	C₂×Dic₆₇	—	2	1	no	—
179	D₂₆₈	—	2	1	no	—
180	D₂₆₈	—	2	1	no	—
181	C₄×D₆₇	—	4	1	no	—
182	C₄×D₆₇	—	4	1	no	—
183	C₄×D₆₇	—	4	2	no	—
184	C₄×D₆₇	—	8	1	no	—
185	Dic₁₃₄	—	2	1	no	—
186	Dic₁₃₄	—	2	1	no	—
187	C₈×C₆₇	—	2	2	no	—
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	4	1	no	—
190	C₈×C₆₇	—	4	1	no	—
191	C₆₇⋊C₈	—	2	2	no	—
192	C₆₇⋊C₈	—	2	2	no	—
193	C₆₇⋊C₈	—	4	1	no	—
194	C₆₇⋊C₈	—	4	2	no	—
195	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
196	C₂²×C₆₇	—	2	1	no	—
197	C₂²×C₆₇	—	2	1	no	—
198	C₂²×C₆₇	#436	1	2	yes	—
199	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×D₆₇	—	4	1	no	—
201	C₂×D₆₇	—	4	1	no	—
202	C₂×D₆₇	—	4	1	no	—
203	C₄×C₆₇	—	2	1	no	—
204	C₄×C₆₇	—	2	1	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	#436	1	2	yes	—
209	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
210	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
211	C₄×C₆₇	#577	1	1	yes	characteristic
212	C₄×C₆₇	#578	1	1	yes	characteristic
213	Dic₆₇	—	4	1	no	—
214	Dic₆₇	—	4	1	no	—
215	Dic₆₇	—	4	1	no	—
216	#17894	—	67	1	no	maximal, Sylow
217	C₂×C₆₇	—	2	2	no	—
218	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
219	C₂×C₆₇	#2947	1	1	yes	characteristic
220	C₂×C₆₇	#2956	1	1	yes	characteristic
221	D₆₇	—	8	1	no	—
222	#2989	—	67	1	no	—
223	#2983	D₆₇	1	1	yes	characteristic, complemented
224	C₄×#137	—	67	1	no	—
225	#1593	—	67	2	no	—
226	#1592	—	67	2	no	—
227	#1591	—	67	2	no	—
228	#1590	—	67	2	no	—
229	#1565	—	67	1	no	—
230	#1476	—	67	1	no	—
231	C₂×Q₈⋊C₈	—	67	1	no	—
232	C₈×M₁₆	—	67	1	no	—
233	C₆₇	#17894	1	1	yes	characteristic, complemented, Sylow
234	C₂×#137	—	67	1	no	—
235	C₄×#41	—	67	1	no	—
236	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
237	#446	—	2	1	no	—
238	#446	—	2	1	no	—
239	#446	—	67	2	no	—
240	#446	C₂×D₆₇	1	2	yes	—
241	#446	C₂×D₆₇	1	2	yes	—
242	#441	—	67	2	no	—
243	#440	—	67	2	no	—
244	C₄×Q₁₆	—	67	2	no	—
245	C₄×SD₁₆	—	67	2	no	—
246	#436	—	67	2	no	—
247	#435	—	134	2	no	—
248	#432	—	2	2	no	—
249	#432	—	134	1	no	—
250	#431	C₂×D₆₇	1	1	yes	characteristic
251	#431	C₂×D₆₇	1	1	yes	characteristic
252	#428	—	67	1	no	—
253	#423	—	134	1	no	—
254	C₂×C₄⋊C₈	—	67	1	no	—
255	#419	—	67	1	no	—
256	#417	—	67	1	no	—
257	C₄×M₁₆	—	67	1	no	—
258	C₂×C₄×C₈	—	67	1	no	—
259	#335	—	67	2	no	—
260	#334	—	67	2	no	—
261	Q₈⋊C₈	—	67	2	no	—
262	Q₈⋊C₈	—	67	2	no	—
263	Q₈⋊C₈	—	67	1	no	—
264	Q₈⋊C₈	—	67	1	no	—
265	D₄⋊C₈	—	67	1	no	—
266	D₄⋊C₈	—	67	1	no	—
267	#322	—	67	2	no	—
268	C₈²	—	67	2	no	—
269	C₂×#41	—	67	1	no	—
270	C₂²×Q₈	C₄×D₆₇	1	1	yes	characteristic
271	#137	—	134	4	no	—
272	C₂×Q₁₆	—	67	2	no	—
273	C₂×SD₁₆	—	67	2	no	—
274	C₂×M₁₆	—	67	1	no	—
275	C₂×M₁₆	—	67	1	no	—
276	C₂×M₁₆	—	134	2	no	—
277	C₂²×C₈	—	134	1	no	—
278	C₂²×C₈	—	134	1	no	—
279	C₄×Q₈	—	2	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×Q₈	—	67	1	no	—
282	C₄×Q₈	—	67	1	no	—
283	C₄×Q₈	C₄×D₆₇	1	2	yes	—
284	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
285	C₄×D₄	—	67	1	no	—
286	C₄×D₄	—	67	1	no	—
287	C₄×D₄	—	134	1	no	—
288	C₄×D₄	—	134	1	no	—
289	#117	—	67	1	no	—
290	#117	—	134	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	—	134	1	no	—
295	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
296	#107	—	67	2	no	—
297	#106	—	67	2	no	—
298	C₄⋊C₈	—	2	2	no	—
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	67	2	no	—
301	C₄⋊C₈	—	134	2	no	—
302	C₄⋊C₈	—	134	1	no	—
303	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
304	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
305	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
306	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
307	#103	—	67	2	no	—
308	#103	—	67	2	no	—
309	#103	—	67	2	no	—
310	#102	—	67	2	no	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₂²⋊C₈	—	134	2	no	—
313	C₂²⋊C₈	C₂²⋊D₆₇	1	2	yes	—
314	#97	—	2	2	no	—
315	#97	—	2	2	no	—
316	#97	—	67	2	no	—
317	#97	—	134	2	no	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	—	67	2	no	—
320	C₄×C₈	—	67	1	no	—
321	C₄×C₈	—	67	1	no	—
322	C₄×C₈	—	134	1	no	—
323	C₄×C₈	C₂²×D₆₇	1	2	yes	—
324	#41	—	134	2	no	—
325	#41	—	134	2	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	67	1	no	—
330	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
331	C₂×D₄	—	67	1	no	—
332	C₂×D₄	—	134	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	134	1	no	—
336	C₂²×C₄	—	134	1	no	—
337	C₂²×C₄	—	134	1	no	—
338	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
339	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
340	C₂²×C₄	#11761628	1	1	yes	characteristic
341	Q₁₆	—	134	4	no	—
342	SD₁₆	—	134	4	no	—
343	M₁₆	—	536	2	no	—
344	M₁₆	—	67	4	no	—
345	M₁₆	—	134	2	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	268	2	no	—
349	C₂×C₈	—	536	1	no	—
350	C₂×C₈	—	67	2	no	—
351	C₂×C₈	—	67	2	no	—
352	C₂×C₈	—	134	2	no	—
353	C₂×C₈	—	134	1	no	—
354	C₂×C₈	—	134	1	no	—
355	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
356	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	67	1	no	—
361	C₄⋊C₄	—	67	1	no	—
362	C₄⋊C₄	—	134	1	no	—
363	C₄⋊C₄	—	134	1	no	—
364	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
365	C₂²⋊C₄	—	134	1	no	—
366	C₂²⋊C₄	—	134	1	no	—
367	C₂²⋊C₄	—	134	1	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	134	1	no	—
371	C₄²	—	134	1	no	—
372	C₄²	—	134	1	no	—
373	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
374	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
375	C₄²	#11761628	1	1	yes	characteristic
376	C₄²	C₂³×D₆₇	1	1	yes	characteristic
377	C₂³	—	134	1	no	—
378	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
379	Q₈	—	2	2	no	—
380	Q₈	—	2	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	134	1	no	—
383	Q₈	—	134	1	no	—
384	D₄	—	268	2	no	—
385	D₄	—	134	1	no	—
386	D₄	—	134	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	8	1	no	—
394	C₂×C₄	—	268	1	no	—
395	C₂×C₄	—	268	1	no	—
396	C₂×C₄	—	268	2	no	—
397	C₂×C₄	—	536	1	no	—
398	C₂×C₄	—	134	1	no	—
399	C₂×C₄	—	134	1	no	—
400	C₂×C₄	—	134	1	no	—
401	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
402	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
403	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
404	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
405	C₂×C₄	#14524051	1	2	yes	—
406	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	2	no	—
411	C₈	—	4	1	no	—
412	C₈	—	4	1	no	—
413	C₈	—	268	1	no	—
414	C₈	—	268	2	no	—
415	C₈	—	67	4	no	—
416	C₈	—	134	2	no	—
417	C₈	—	134	2	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	268	1	no	—
421	C₂²	—	268	1	no	—
422	C₂²	—	268	1	no	—
423	C₂²	#16863472	1	2	yes	—
424	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	268	1	no	—
431	C₄	—	268	1	no	—
432	C₄	—	268	1	no	—
433	C₄	#16863472	1	2	yes	—
434	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
435	C₄	C₂×#14524051	1	1	yes	characteristic
436	C₄	#16864204	1	1	yes	characteristic
437	C₄	#16864211	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	536	1	no	—
440	C₂	C₂×#16863472	1	1	yes	characteristic
441	C₂	#22569052	1	1	yes	characteristic
442	C₂	#22569116	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#2983	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#2983	D₆₇
hypocenter	C₆₇	#17894
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2983

Complements

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

N	H ≅ G/N	type
3. #22559116	C₂	semidirect
4. #22559109	C₂	semidirect
5. #22559104	C₂	semidirect
6. #22559099	C₂	semidirect
7. #22558962	C₂	semidirect
9. C₂×Q₈⋊(C₆₇⋊C₈)	C₂	semidirect
10. #22558490	C₂	semidirect
11. C₆₇×#2983	C₂	semidirect
113. #11761612	M₁₆	semidirect
223. #2983	D₆₇	semidirect
233. C₆₇	#17894	semidirect

Maximal subgroups

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

subgroup	index
#22569322	2
C₄×#14524074	2
#22559116	2
#22559109	2
#22559104	2
#22559099	2
#22558962	2
#22558589	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
#22558490	2
C₆₇×#2983	2
#17894	67