Subgroups — #30827276

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569229	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569221	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
4	#22559102	C₂	1	2	yes	maximal, complemented
5	#22559056	C₂	1	2	yes	maximal, complemented
6	#22559041	C₂	1	2	yes	maximal, complemented
7	#22559006	C₂	1	2	yes	maximal, complemented
8	#22558970	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558644	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1488	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₄	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863523	C₂²	1	2	yes	—
15	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
16	C₄×#11761618	C₄	1	1	yes	characteristic
17	C₄×#11761618	C₂²	1	1	yes	characteristic
18	C₄×#11761615	C₄	1	1	yes	characteristic
19	C₄×#11761615	C₂²	1	1	yes	characteristic
20	C₄×#11761616	C₄	1	1	yes	characteristic
21	C₄×#11761616	C₂²	1	1	yes	characteristic
22	C₄×#11761613	C₄	1	1	yes	characteristic
23	C₄×#11761613	C₂²	1	1	yes	characteristic
24	#16863474	—	2	2	no	—
25	#16863474	C₂²	1	2	yes	—
26	#16863470	—	2	2	no	—
27	#16863461	C₂²	1	2	yes	—
28	#16863453	—	2	1	no	—
29	#16863453	C₂²	1	1	yes	characteristic
30	#16863451	—	2	1	no	—
31	#16863432	C₄	1	1	yes	characteristic
32	#16863392	C₄	1	1	yes	characteristic
33	#16863390	C₄	1	1	yes	characteristic
34	C₄×#11761612	—	2	1	no	—
35	C₄×#11761612	C₂²	1	1	yes	characteristic
36	#97×D₆₇	—	2	1	no	—
37	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
38	#16863148	C₂²	1	2	yes	—
39	#16863146	C₂²	1	2	yes	—
40	#16863128	C₂²	1	2	yes	—
41	#16863125	C₂²	1	2	yes	—
42	#16863118	C₂²	1	2	yes	—
43	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
44	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
45	#432×C₆₇	C₂²	1	1	yes	characteristic
46	#431×C₆₇	C₂²	1	1	yes	characteristic
47	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
48	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	#14524063	—	2	4	no	—
51	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
52	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761612	—	2	2	no	—
56	C₂×#11761612	D₄	1	2	yes	—
57	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₆₇	—	2	2	no	—
60	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
62	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
63	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
64	C₄×C₂²⋊D₆₇	—	2	1	no	—
65	C₄×C₂²⋊D₆₇	—	2	1	no	—
66	#14524002	C₂×C₄	1	1	yes	characteristic
67	#14524000	—	2	1	no	—
68	C₄²×D₆₇	—	2	1	no	—
69	#14523976	C₂×C₄	1	1	yes	characteristic
70	#14523976	C₂×C₄	1	1	yes	characteristic
71	#14523974	C₂×C₄	1	1	yes	characteristic
72	#14523974	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
74	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
75	Dic₆₇⋊C₈	—	2	2	no	—
76	#14523967	C₂×C₄	1	1	yes	characteristic
77	#14523967	C₂×C₄	1	1	yes	characteristic
78	#14523966	C₂×C₄	1	1	yes	characteristic
79	#14523966	C₂×C₄	1	1	yes	characteristic
80	#14523964	C₂×C₄	1	1	yes	characteristic
81	#14523964	C₂×C₄	1	1	yes	characteristic
82	#14523963	C₂×C₄	1	1	yes	characteristic
83	#14523963	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
85	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
86	(C₂×D₆₇)⋊C₈	—	2	2	no	—
87	#14523954	—	2	1	no	—
88	#14523954	D₄	1	1	yes	characteristic
89	#14523954	D₄	1	1	yes	characteristic
90	#14523953	—	2	2	no	—
91	C₄×C₆₇⋊C₈	—	2	2	no	—
92	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
93	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
96	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
97	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₆₇	—	2	1	no	—
101	C₄×D₄×C₆₇	—	2	1	no	—
102	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
104	#117×C₆₇	—	2	1	no	—
105	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	—	2	1	no	—
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	C₄⋊C₈×C₆₇	—	2	1	no	—
109	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
110	C₂²⋊C₈×C₆₇	—	2	1	no	—
111	C₂²⋊C₈×C₆₇	—	2	1	no	—
112	#97×C₆₇	—	2	1	no	—
113	C₄×C₈×C₆₇	C₂³	1	2	yes	—
114	#11761628	—	2	2	no	—
115	#11761628	—	4	1	no	—
116	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
117	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₂²⋊D₆₇	—	2	1	no	—
119	C₂×C₄×D₆₇	—	2	1	no	—
120	C₂×C₄×D₆₇	—	2	1	no	—
121	C₂×C₄×D₆₇	—	2	1	no	—
122	#11761618	—	2	2	no	—
123	#11761616	—	2	2	no	—
124	#11761615	M₁₆	1	4	yes	complemented
125	#11761613	—	2	2	no	—
126	#11761612	—	2	4	no	—
127	#11761612	—	4	4	no	—
128	C₂×C₆₇⋊C₈	—	2	2	no	—
129	C₂×C₆₇⋊C₈	—	2	2	no	—
130	C₂×C₆₇⋊C₈	—	4	1	no	—
131	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
136	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
137	C₈×D₆₇	—	4	4	no	—
138	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
140	Dic₆₇⋊C₄	—	2	1	no	—
141	Dic₆₇⋊C₄	—	2	1	no	—
142	C₂²⋊Dic₆₇	—	2	1	no	—
143	(C₂×D₆₇)⋊C₄	—	2	1	no	—
144	(C₂×D₆₇)⋊C₄	—	2	1	no	—
145	C₄×Dic₆₇	—	2	1	no	—
146	C₄×Dic₆₇	—	2	1	no	—
147	C₄×Dic₆₇	—	2	1	no	—
148	#41×C₆₇	—	2	2	no	—
149	#41×C₆₇	—	4	1	no	—
150	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	1	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂²⋊C₄×C₆₇	—	2	1	no	—
166	C₂²⋊C₄×C₆₇	—	2	1	no	—
167	C₂²⋊C₄×C₆₇	—	2	1	no	—
168	C₄²×C₆₇	—	2	1	no	—
169	C₄²×C₆₇	—	2	1	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₆₇	#41	1	1	yes	characteristic
174	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₂³×C₆₇	—	2	1	no	—
176	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₆₇	—	2	1	no	—
178	Q₈×C₆₇	—	2	1	no	—
179	Q₈×C₆₇	C₂×M₁₆	1	2	yes	—
180	D₄×C₆₇	—	2	1	no	—
181	D₄×C₆₇	—	2	1	no	—
182	D₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	8	1	no	—
193	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
199	C₂²⋊D₆₇	—	4	2	no	—
200	C₂×Dic₆₇	—	2	1	no	—
201	C₂×Dic₆₇	—	2	1	no	—
202	C₂×Dic₆₇	—	2	1	no	—
203	D₂₆₈	—	2	1	no	—
204	D₂₆₈	C₂×M₁₆	1	2	yes	—
205	C₄×D₆₇	—	4	2	no	—
206	C₄×D₆₇	—	4	2	no	—
207	C₄×D₆₇	—	8	1	no	—
208	Dic₁₃₄	—	2	1	no	—
209	Dic₁₃₄	—	2	1	no	—
210	C₈×C₆₇	—	4	2	no	—
211	C₈×C₆₇	—	4	2	no	—
212	C₆₇⋊C₈	—	2	1	no	—
213	C₆₇⋊C₈	—	2	1	no	—
214	C₆₇⋊C₈	—	2	1	no	—
215	C₆₇⋊C₈	—	2	2	no	—
216	C₆₇⋊C₈	—	4	2	no	—
217	C₆₇⋊C₈	C₂×M₁₆	1	2	yes	—
218	C₆₇⋊C₈	#135	1	4	yes	complemented
219	C₂²×C₆₇	—	2	1	no	—
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	1	no	—
222	C₂²×C₆₇	—	4	1	no	—
223	C₂²×C₆₇	—	4	1	no	—
224	C₂²×C₆₇	—	4	1	no	—
225	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂×D₆₇	—	2	2	no	—
227	C₂×D₆₇	—	4	1	no	—
228	C₂×D₆₇	—	4	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	4	1	no	—
235	C₄×C₆₇	—	4	1	no	—
236	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
237	C₄×C₆₇	#568	1	1	yes	characteristic
238	C₄×C₆₇	C₂×#135	1	2	yes	—
239	Dic₆₇	—	4	1	no	—
240	Dic₆₇	—	4	1	no	—
241	Dic₆₇	—	4	1	no	—
242	#18037	—	67	1	no	maximal, Sylow
243	C₂×C₆₇	—	4	1	no	—
244	C₂×C₆₇	—	8	1	no	—
245	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
246	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
247	C₂×C₆₇	#2957	1	1	yes	characteristic
248	D₆₇	—	4	2	no	—
249	#2971	—	67	1	no	—
250	#2970	—	67	1	no	—
251	C₄×#135	—	67	1	no	—
252	#1590	—	67	2	no	—
253	#1584	—	67	2	no	—
254	#1581	—	67	2	no	—
255	C₈×SD₁₆	—	67	2	no	—
256	#1568	—	67	1	no	—
257	#1488	—	67	1	no	—
258	#1488	D₆₇	1	1	yes	characteristic, complemented
259	C₈×M₁₆	—	67	1	no	—
260	C₆₇	#18037	1	1	yes	characteristic, complemented, Sylow
261	C₂×#135	—	67	1	no	—
262	C₄×#41	—	67	1	no	—
263	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
264	#446	—	67	2	no	—
265	C₈×Q₈	—	67	2	no	—
266	C₄×Q₁₆	—	67	1	no	—
267	C₄×Q₁₆	—	67	1	no	—
268	C₄×SD₁₆	—	67	1	no	—
269	C₄×SD₁₆	—	67	1	no	—
270	C₄×SD₁₆	—	67	1	no	—
271	C₄×SD₁₆	—	67	1	no	—
272	C₄×D₈	—	67	1	no	—
273	C₄×D₈	—	67	1	no	—
274	#436	—	67	2	no	—
275	#436	—	134	2	no	—
276	#435	—	134	2	no	—
277	C₈×D₄	—	67	2	no	—
278	#432	—	67	1	no	—
279	#432	—	134	1	no	—
280	#432	—	134	1	no	—
281	#432	C₂×D₆₇	1	1	yes	characteristic
282	#431	C₂×D₆₇	1	1	yes	characteristic
283	#427	—	67	1	no	—
284	#416	—	67	1	no	—
285	#416	—	67	1	no	—
286	C₄×M₁₆	—	67	1	no	—
287	C₄×M₁₆	—	134	1	no	—
288	C₂×#97	—	134	1	no	—
289	C₂×C₄×C₈	—	67	1	no	—
290	#335	—	67	2	no	—
291	#334	—	67	2	no	—
292	Q₈⋊C₈	—	67	2	no	—
293	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
294	D₄⋊C₈	—	67	2	no	—
295	D₄⋊C₈	C₂×D₆₇	1	2	yes	—
296	#322	—	67	2	no	—
297	C₈²	—	67	2	no	—
298	C₂×#41	—	67	1	no	—
299	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
300	#135	—	134	4	no	—
301	C₂×Q₁₆	—	67	1	no	—
302	C₂×SD₁₆	—	67	1	no	—
303	C₂×SD₁₆	—	67	1	no	—
304	C₂×D₈	—	67	1	no	—
305	C₂×M₁₆	—	67	2	no	—
306	C₂×M₁₆	—	134	2	no	—
307	C₂²×C₈	—	67	1	no	—
308	C₂²×C₈	—	67	1	no	—
309	C₂²×C₈	—	134	2	no	—
310	C₄×Q₈	—	67	1	no	—
311	C₄×Q₈	—	67	1	no	—
312	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
313	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	67	1	no	—
317	C₄×D₄	—	67	1	no	—
318	C₄×D₄	—	134	1	no	—
319	C₄×D₄	—	134	1	no	—
320	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
321	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	—	67	1	no	—
324	#117	—	134	1	no	—
325	#117	C₄×D₆₇	1	1	yes	characteristic
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	—	134	1	no	—
328	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
329	#107	—	67	1	no	—
330	#107	—	67	1	no	—
331	#106	—	67	1	no	—
332	#106	—	67	1	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	67	2	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
338	#103	—	67	1	no	—
339	#103	—	67	1	no	—
340	#103	—	67	1	no	—
341	#103	—	67	1	no	—
342	#102	—	67	1	no	—
343	#102	—	67	1	no	—
344	#102	—	67	1	no	—
345	#102	—	67	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	C₂²⋊C₈	—	2	1	no	—
348	C₂²⋊C₈	—	134	2	no	—
349	C₂²⋊C₈	—	134	2	no	—
350	C₂²⋊C₈	—	134	1	no	—
351	#97	—	2	1	no	—
352	#97	—	67	1	no	—
353	#97	—	67	1	no	—
354	#97	—	134	2	no	—
355	#97	—	134	1	no	—
356	C₄×C₈	—	67	1	no	—
357	C₄×C₈	—	67	1	no	—
358	C₄×C₈	—	67	1	no	—
359	C₄×C₈	—	67	2	no	—
360	C₄×C₈	—	67	1	no	—
361	C₄×C₈	—	134	2	no	—
362	C₄×C₈	C₂²×D₆₇	1	2	yes	—
363	#41	—	2	2	no	—
364	#41	—	4	1	no	—
365	#41	—	268	1	no	—
366	#41	—	134	2	no	—
367	C₂×Q₈	—	67	1	no	—
368	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	67	1	no	—
371	C₂×D₄	—	134	1	no	—
372	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	134	1	no	—
377	C₂²×C₄	—	134	1	no	—
378	C₂²×C₄	—	134	1	no	—
379	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
380	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
381	C₂²×C₄	#11761627	1	1	yes	characteristic
382	Q₁₆	—	134	2	no	—
383	SD₁₆	—	67	4	no	—
384	SD₁₆	—	134	2	no	—
385	D₈	—	134	2	no	—
386	M₁₆	—	268	4	no	—
387	M₁₆	—	134	4	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	268	4	no	—
391	C₂×C₈	—	268	1	no	—
392	C₂×C₈	—	67	1	no	—
393	C₂×C₈	—	67	2	no	—
394	C₂×C₈	—	67	1	no	—
395	C₂×C₈	—	67	1	no	—
396	C₂×C₈	—	67	1	no	—
397	C₂×C₈	—	67	2	no	—
398	C₂×C₈	—	134	2	no	—
399	C₂×C₈	—	134	2	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	67	1	no	—
403	C₄⋊C₄	—	67	1	no	—
404	C₄⋊C₄	—	134	1	no	—
405	C₄⋊C₄	—	134	1	no	—
406	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
407	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
408	C₂²⋊C₄	—	2	1	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	134	1	no	—
412	C₂²⋊C₄	—	134	1	no	—
413	C₂²⋊C₄	—	134	1	no	—
414	C₄²	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	134	1	no	—
418	C₄²	—	134	1	no	—
419	C₄²	—	134	1	no	—
420	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
421	C₄²	D₄×D₆₇	1	1	yes	characteristic
422	C₄²	#11761627	1	1	yes	characteristic
423	C₄²	C₂³×D₆₇	1	1	yes	characteristic
424	C₂³	—	2	1	no	—
425	C₂³	—	134	1	no	—
426	C₂³	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
427	Q₈	—	2	1	no	—
428	Q₈	—	134	1	no	—
429	Q₈	—	134	1	no	—
430	Q₈	M₁₆×D₆₇	1	2	yes	—
431	D₄	—	2	1	no	—
432	D₄	—	2	1	no	—
433	D₄	—	4	2	no	—
434	D₄	—	268	2	no	—
435	D₄	—	67	2	no	—
436	D₄	—	134	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	8	1	no	—
447	C₂×C₄	—	268	2	no	—
448	C₂×C₄	—	268	2	no	—
449	C₂×C₄	—	536	1	no	—
450	C₂×C₄	—	134	1	no	—
451	C₂×C₄	—	134	1	no	—
452	C₂×C₄	—	134	1	no	—
453	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
454	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
455	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
456	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
457	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
458	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
459	C₈	—	4	2	no	—
460	C₈	—	4	2	no	—
461	C₈	—	268	2	no	—
462	C₈	—	67	2	no	—
463	C₈	—	67	4	no	—
464	C₈	—	134	1	no	—
465	C₈	—	134	1	no	—
466	C₈	—	134	1	no	—
467	C₈	—	134	2	no	—
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	268	1	no	—
475	C₂²	—	268	1	no	—
476	C₂²	—	134	2	no	—
477	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	268	1	no	—
486	C₄	—	268	1	no	—
487	C₄	—	268	1	no	—
488	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
489	C₄	#16864162	1	1	yes	characteristic
490	C₄	#135×D₆₇	1	2	yes	—
491	C₂	—	4	1	no	—
492	C₂	—	8	1	no	—
493	C₂	—	268	2	no	—
494	C₂	#22568941	1	1	yes	characteristic
495	C₂	#22568977	1	1	yes	characteristic
496	C₂	#22569122	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1488	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1488	D₆₇
hypocenter	C₆₇	#18037
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1488

Complements

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

N	H ≅ G/N	type
1. #22569229	C₂	semidirect
2. #22569221	C₂	semidirect
4. #22559102	C₂	semidirect
5. #22559056	C₂	semidirect
6. #22559041	C₂	semidirect
7. #22559006	C₂	semidirect
8. #22558970	C₂	semidirect
9. #22558644	C₂	semidirect
10. #22558490	C₂	semidirect
11. C₆₇×#1488	C₂	semidirect
124. #11761615	M₁₆	semidirect
218. C₆₇⋊C₈	#135	semidirect
258. #1488	D₆₇	semidirect
260. C₆₇	#18037	semidirect

Maximal subgroups

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

subgroup	index
#22569229	2
#22569221	2
C₄×#14524063	2
#22559102	2
#22559056	2
#22559041	2
#22559006	2
#22558970	2
#22558644	2
#22558490	2
C₆₇×#1488	2
#18037	67