Subgroups — #30875537

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
2	#22568874	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561101	C₂	1	2	yes	maximal
4	#22560857	C₂	1	2	yes	maximal, complemented
5	#22560854	C₂	1	2	yes	maximal, complemented
6	#22560380	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560229	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560226	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560094	C₂	1	2	yes	maximal, complemented
10	#22560079	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2925	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
15	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
16	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
17	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
18	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
19	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
20	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
21	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
22	#16863456	—	2	2	no	—
23	#16863452	—	2	1	no	—
24	#16863448	—	2	1	no	—
25	C₂×#14523978	C₂²	1	2	yes	—
26	C₂×#14523978	C₂²	1	2	yes	—
27	#16863432	C₂²	1	1	yes	characteristic
28	#16863420	C₂²	1	1	yes	characteristic
29	C₂×#14523970	C₂²	1	2	yes	—
30	C₂×#14523969	C₂²	1	2	yes	—
31	#16863392	C₂²	1	1	yes	characteristic
32	#16863390	C₂²	1	1	yes	characteristic
33	#16863362	C₂²	1	2	yes	—
34	#16863351	C₂²	1	2	yes	—
35	C₂×#14523953	C₂²	1	2	yes	—
36	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
38	#16863157	C₂²	1	2	yes	—
39	#16863155	C₂²	1	2	yes	—
40	#16863150	C₂²	1	2	yes	—
41	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
42	#433×C₆₇	—	2	2	no	—
43	#432×C₆₇	—	2	1	no	—
44	#431×C₆₇	—	2	1	no	—
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
47	#408×C₆₇	C₂²	1	2	yes	—
48	#405×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂³	1	1	yes	characteristic
50	#14524063	—	2	2	no	—
51	#14524063	—	2	2	no	—
52	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
56	C₂×#11761612	C₂³	1	2	yes	—
57	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
59	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
61	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
62	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₂²⋊D₆₇	—	2	1	no	—
64	C₄×C₂²⋊D₆₇	—	2	1	no	—
65	#14524002	C₂³	1	1	yes	characteristic
66	#14524000	—	2	1	no	—
67	C₂×C₄⋊Dic₆₇	—	2	1	no	—
68	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
69	C₄²×D₆₇	—	2	1	no	—
70	#14523978	—	2	4	no	—
71	#14523978	—	4	2	no	—
72	#14523976	C₂×C₄	1	1	yes	characteristic
73	#14523976	C₂×C₄	1	1	yes	characteristic
74	#14523974	C₂×C₄	1	1	yes	characteristic
75	#14523974	C₂×C₄	1	1	yes	characteristic
76	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
77	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
78	#14523970	—	4	2	no	—
79	#14523969	—	4	2	no	—
80	#14523967	C₂×C₄	1	1	yes	characteristic
81	#14523967	C₂×C₄	1	1	yes	characteristic
82	#14523966	C₂×C₄	1	1	yes	characteristic
83	#14523966	C₂×C₄	1	1	yes	characteristic
84	#14523964	C₂×C₄	1	1	yes	characteristic
85	#14523964	C₂×C₄	1	1	yes	characteristic
86	#14523963	C₂×C₄	1	1	yes	characteristic
87	#14523963	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₆₇)⋊C₈	—	2	2	no	—
89	(C₂×D₆₇)⋊C₈	—	2	2	no	—
90	#14523953	—	2	4	no	—
91	#14523953	—	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	C₂×C₄	1	1	yes	characteristic
95	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₈×Dic₆₇	—	2	2	no	—
97	C₆₇⋊#95	—	2	1	no	—
98	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
100	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
101	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₄×D₄×C₆₇	—	2	1	no	—
104	C₄×D₄×C₆₇	—	2	1	no	—
105	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
107	#117×C₆₇	—	2	1	no	—
108	#117×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₆₇	—	2	1	no	—
112	C₄⋊C₈×C₆₇	—	2	1	no	—
113	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
115	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
116	C₂²⋊C₈×C₆₇	—	2	2	no	—
117	C₂²⋊C₈×C₆₇	—	2	2	no	—
118	#97×C₆₇	—	2	2	no	—
119	C₄×C₈×C₆₇	—	2	2	no	—
120	#11761628	—	2	1	no	—
121	#11761628	—	2	1	no	—
122	#11761628	—	2	1	no	—
123	#11761628	—	2	1	no	—
124	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₂²⋊D₆₇	—	2	1	no	—
127	C₂²×Dic₆₇	—	2	2	no	—
128	C₂×C₄×D₆₇	—	2	1	no	—
129	C₂×C₄×D₆₇	—	2	1	no	—
130	C₂×C₄×D₆₇	—	2	1	no	—
131	#11761618	—	2	2	no	—
132	#11761616	—	2	2	no	—
133	#11761615	—	2	2	no	—
134	#11761613	—	2	2	no	—
135	#11761612	—	2	2	no	—
136	#11761612	—	2	2	no	—
137	C₂×C₆₇⋊C₈	—	2	1	no	—
138	C₂×C₆₇⋊C₈	—	2	1	no	—
139	C₂×C₆₇⋊C₈	—	2	2	no	—
140	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₆₇⋊C₈	#41	1	2	yes	—
145	C₂×C₆₇⋊C₈	#41	1	2	yes	—
146	C₄⋊Dic₆₇	—	4	2	no	—
147	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
149	Dic₆₇⋊C₄	—	2	1	no	—
150	Dic₆₇⋊C₄	—	2	1	no	—
151	C₂²⋊Dic₆₇	—	2	1	no	—
152	(C₂×D₆₇)⋊C₄	—	2	1	no	—
153	(C₂×D₆₇)⋊C₄	—	2	1	no	—
154	C₄×Dic₆₇	—	2	1	no	—
155	C₄×Dic₆₇	—	2	1	no	—
156	C₄×Dic₆₇	—	2	1	no	—
157	C₄×Dic₆₇	—	2	2	no	—
158	C₄×Dic₆₇	—	2	2	no	—
159	#41×C₆₇	—	2	1	no	—
160	#41×C₆₇	—	2	1	no	—
161	#41×C₆₇	—	2	1	no	—
162	#41×C₆₇	—	2	1	no	—
163	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂×D₄×C₆₇	—	2	1	no	—
165	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	1	no	—
169	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	M₁₆×C₆₇	—	2	2	no	—
173	M₁₆×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₂×C₈×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
178	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²⋊C₄×C₆₇	—	2	1	no	—
184	C₂²⋊C₄×C₆₇	—	2	1	no	—
185	C₂²⋊C₄×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	2	1	no	—
188	C₄²×C₆₇	—	2	1	no	—
189	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
190	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
195	C₂²×D₆₇	—	2	1	no	—
196	Q₈×C₆₇	—	2	1	no	—
197	Q₈×C₆₇	—	2	1	no	—
198	D₄×C₆₇	—	2	1	no	—
199	D₄×C₆₇	—	2	1	no	—
200	D₄×C₆₇	—	4	1	no	—
201	D₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	8	1	no	—
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	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
221	C₂²⋊D₆₇	—	4	1	no	—
222	C₂²⋊D₆₇	—	4	1	no	—
223	C₂×Dic₆₇	—	2	1	no	—
224	C₂×Dic₆₇	—	2	1	no	—
225	C₂×Dic₆₇	—	2	1	no	—
226	C₂×Dic₆₇	—	4	2	no	—
227	C₂×Dic₆₇	—	4	2	no	—
228	C₂×Dic₆₇	—	4	2	no	—
229	D₂₆₈	—	2	1	no	—
230	D₂₆₈	—	2	1	no	—
231	C₄×D₆₇	—	4	1	no	—
232	C₄×D₆₇	—	4	1	no	—
233	C₄×D₆₇	—	4	2	no	—
234	C₄×D₆₇	—	8	1	no	—
235	Dic₁₃₄	—	2	1	no	—
236	Dic₁₃₄	—	2	1	no	—
237	C₈×C₆₇	—	2	2	no	—
238	C₈×C₆₇	—	2	2	no	—
239	C₈×C₆₇	—	4	2	no	—
240	C₆₇⋊C₈	—	2	1	no	—
241	C₆₇⋊C₈	—	2	1	no	—
242	C₆₇⋊C₈	—	4	2	no	—
243	C₆₇⋊C₈	#117	1	2	yes	—
244	C₆₇⋊C₈	#117	1	2	yes	—
245	C₂²×C₆₇	—	2	1	no	—
246	C₂²×C₆₇	—	2	1	no	—
247	C₂²×C₆₇	—	4	1	no	—
248	C₂²×C₆₇	—	4	1	no	—
249	C₂²×C₆₇	—	4	1	no	—
250	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂²×C₆₇	#555	1	1	yes	characteristic
252	C₂²×C₆₇	#556	1	1	yes	characteristic
253	C₂×D₆₇	—	4	1	no	—
254	C₂×D₆₇	—	4	1	no	—
255	C₂×D₆₇	—	4	1	no	—
256	C₄×C₆₇	—	2	1	no	—
257	C₄×C₆₇	—	2	1	no	—
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	4	1	no	—
260	C₄×C₆₇	—	4	1	no	—
261	C₄×C₆₇	—	4	1	no	—
262	C₄×C₆₇	#444	1	2	yes	—
263	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
264	C₄×C₆₇	#520	1	1	yes	characteristic
265	C₄×C₆₇	#547	1	1	yes	characteristic
266	C₄×C₆₇	#550	1	1	yes	characteristic
267	Dic₆₇	—	4	1	no	—
268	Dic₆₇	—	4	1	no	—
269	Dic₆₇	—	4	1	no	—
270	Dic₆₇	—	8	2	no	—
271	#22803	—	67	1	no	maximal, Sylow
272	C₂×C₆₇	—	2	1	no	—
273	C₂×C₆₇	—	2	1	no	—
274	C₂×C₆₇	—	8	1	no	—
275	C₂×C₆₇	#2354	1	1	yes	characteristic
276	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
277	C₂×C₆₇	#2957	1	1	yes	characteristic
278	D₆₇	—	8	1	no	—
279	C₄×#135	—	67	1	no	—
280	#2925	—	67	1	no	—
281	#2925	D₆₇	1	1	yes	characteristic, complemented
282	#1975	—	67	2	no	—
283	#1929	—	67	2	no	—
284	#1929	—	67	2	no	—
285	#1848	—	67	1	no	—
286	#1806	—	67	1	no	—
287	#1806	—	67	1	no	—
288	#1773	—	67	2	no	—
289	#1770	—	67	1	no	—
290	C₆₇	#22803	1	1	yes	characteristic, complemented, Sylow
291	C₂×#135	—	67	1	no	—
292	C₄×#41	—	67	1	no	—
293	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
294	C₄×Q₁₆	—	67	1	no	—
295	C₄×Q₁₆	—	67	1	no	—
296	C₄×SD₁₆	—	67	1	no	—
297	C₄×SD₁₆	—	67	1	no	—
298	C₄×SD₁₆	—	67	1	no	—
299	C₄×SD₁₆	—	67	1	no	—
300	C₄×D₈	—	67	1	no	—
301	C₄×D₈	—	67	1	no	—
302	#433	—	2	2	no	—
303	#433	—	134	2	no	—
304	#432	—	2	1	no	—
305	#432	—	134	1	no	—
306	#431	—	2	1	no	—
307	#431	—	134	1	no	—
308	C₂×#108	—	67	2	no	—
309	C₂×#108	—	67	2	no	—
310	#427	—	67	1	no	—
311	#423	—	67	1	no	—
312	#423	C₂×D₆₇	1	1	yes	characteristic
313	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
314	C₂×#104	—	67	2	no	—
315	C₂×#104	—	67	2	no	—
316	#416	—	67	1	no	—
317	#416	—	67	1	no	—
318	#408	—	67	2	no	—
319	#408	C₂×D₆₇	1	2	yes	—
320	#405	—	67	2	no	—
321	#405	C₂×D₆₇	1	2	yes	—
322	C₂×#97	—	67	2	no	—
323	C₂×C₄×C₈	—	67	1	no	—
324	C₄×C₄⋊C₄	—	67	1	no	—
325	#339	—	67	2	no	—
326	#339	—	67	2	no	—
327	#336	—	67	2	no	—
328	C₂×#41	—	67	1	no	—
329	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
330	#135	—	134	2	no	—
331	#135	—	134	2	no	—
332	C₂×Q₁₆	—	67	1	no	—
333	C₂×SD₁₆	—	67	1	no	—
334	C₂×SD₁₆	—	67	1	no	—
335	C₂×D₈	—	67	1	no	—
336	C₂×M₁₆	—	67	2	no	—
337	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
338	C₂²×C₈	—	67	1	no	—
339	C₂²×C₈	—	67	1	no	—
340	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
341	C₄×Q₈	—	67	1	no	—
342	C₄×Q₈	—	67	1	no	—
343	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
344	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	67	1	no	—
348	C₄×D₄	—	67	1	no	—
349	C₄×D₄	—	134	1	no	—
350	C₄×D₄	—	134	1	no	—
351	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
352	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
353	#117	—	2	1	no	—
354	#117	—	67	1	no	—
355	#117	—	134	1	no	—
356	#117	C₂²×D₆₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	—	134	1	no	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	—	67	2	no	—
360	C₂×C₄²	—	134	1	no	—
361	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
362	#108	—	268	2	no	—
363	#108	—	134	4	no	—
364	#107	—	67	1	no	—
365	#107	—	67	1	no	—
366	#106	—	67	1	no	—
367	#106	—	67	1	no	—
368	C₄⋊C₈	—	2	1	no	—
369	C₄⋊C₈	—	2	1	no	—
370	C₄⋊C₈	—	134	1	no	—
371	C₄⋊C₈	—	134	1	no	—
372	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
373	C₄⋊C₈	C₂²⋊D₆₇	1	1	yes	characteristic
374	C₄⋊C₈	C₂²⋊D₆₇	1	2	yes	—
375	#104	—	268	2	no	—
376	#104	—	268	2	no	—
377	#103	—	67	1	no	—
378	#103	—	67	1	no	—
379	#103	—	67	1	no	—
380	#103	—	67	1	no	—
381	#102	—	67	1	no	—
382	#102	—	67	1	no	—
383	#102	—	67	1	no	—
384	#102	—	67	1	no	—
385	C₂²⋊C₈	—	2	2	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	C₂²⋊C₈	—	134	2	no	—
388	C₂²⋊C₈	—	134	2	no	—
389	#97	—	2	2	no	—
390	#97	—	134	4	no	—
391	#97	—	134	2	no	—
392	C₄×C₈	—	2	2	no	—
393	C₄×C₈	—	67	1	no	—
394	C₄×C₈	—	67	1	no	—
395	C₄×C₈	—	67	1	no	—
396	C₄×C₈	—	67	1	no	—
397	C₄×C₈	—	134	2	no	—
398	#95	—	134	1	no	—
399	#41	—	2	1	no	—
400	#41	—	2	1	no	—
401	#41	—	2	1	no	—
402	#41	—	2	1	no	—
403	#41	—	134	1	no	—
404	#41	—	134	1	no	—
405	#41	—	134	1	no	—
406	#41	—	134	1	no	—
407	C₂×Q₈	—	67	1	no	—
408	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	67	1	no	—
411	C₂×D₄	—	134	1	no	—
412	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	134	2	no	—
417	C₂²×C₄	—	134	1	no	—
418	C₂²×C₄	—	134	1	no	—
419	C₂²×C₄	—	134	1	no	—
420	C₂²×C₄	#11761628	1	1	yes	characteristic
421	C₂²×C₄	#11761628	1	1	yes	characteristic
422	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
423	Q₁₆	—	134	2	no	—
424	SD₁₆	—	134	2	no	—
425	SD₁₆	—	134	2	no	—
426	D₈	—	134	2	no	—
427	M₁₆	—	2	2	no	—
428	M₁₆	—	2	2	no	—
429	M₁₆	—	134	2	no	—
430	M₁₆	—	134	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	67	1	no	—
435	C₂×C₈	—	67	1	no	—
436	C₂×C₈	—	67	2	no	—
437	C₂×C₈	—	67	2	no	—
438	C₂×C₈	—	67	1	no	—
439	C₂×C₈	—	67	1	no	—
440	C₂×C₈	—	134	2	no	—
441	C₂×C₈	—	134	1	no	—
442	C₂×C₈	—	134	1	no	—
443	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
444	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	268	2	no	—
448	C₄⋊C₄	—	67	1	no	—
449	C₄⋊C₄	—	67	1	no	—
450	C₄⋊C₄	—	134	1	no	—
451	C₄⋊C₄	—	134	1	no	—
452	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
453	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
454	C₂²⋊C₄	—	2	1	no	—
455	C₂²⋊C₄	—	2	1	no	—
456	C₂²⋊C₄	—	2	1	no	—
457	C₂²⋊C₄	—	134	1	no	—
458	C₂²⋊C₄	—	134	1	no	—
459	C₂²⋊C₄	—	134	1	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	2	1	no	—
462	C₄²	—	2	1	no	—
463	C₄²	—	134	2	no	—
464	C₄²	—	134	2	no	—
465	C₄²	—	134	1	no	—
466	C₄²	—	134	1	no	—
467	C₄²	—	134	1	no	—
468	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
469	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
470	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
471	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
472	C₂³	—	2	1	no	—
473	C₂³	—	134	1	no	—
474	C₂³	C₂×#11761628	1	1	yes	characteristic
475	Q₈	—	2	1	no	—
476	Q₈	—	2	1	no	—
477	Q₈	—	134	1	no	—
478	Q₈	—	134	1	no	—
479	D₄	—	2	1	no	—
480	D₄	—	2	1	no	—
481	D₄	—	4	1	no	—
482	D₄	—	4	1	no	—
483	D₄	—	268	1	no	—
484	D₄	—	268	1	no	—
485	D₄	—	134	1	no	—
486	D₄	—	134	1	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	8	1	no	—
496	C₂×C₄	—	268	2	no	—
497	C₂×C₄	—	268	1	no	—
498	C₂×C₄	—	268	1	no	—
499	C₂×C₄	—	268	2	no	—
500	C₂×C₄	—	268	2	no	—
501	C₂×C₄	—	268	2	no	—
502	C₂×C₄	—	536	1	no	—
503	C₂×C₄	—	134	1	no	—
504	C₂×C₄	—	134	1	no	—
505	C₂×C₄	—	134	1	no	—
506	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
507	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
508	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
509	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
512	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
513	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
514	C₂×C₄	#14524089	1	1	yes	characteristic
515	C₂×C₄	#14524090	1	1	yes	characteristic
516	C₈	—	2	2	no	—
517	C₈	—	2	2	no	—
518	C₈	—	4	2	no	—
519	C₈	—	268	2	no	—
520	C₈	—	67	2	no	—
521	C₈	—	67	2	no	—
522	C₈	—	134	1	no	—
523	C₈	—	134	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	4	1	no	—
527	C₂²	—	4	1	no	—
528	C₂²	—	4	1	no	—
529	C₂²	—	268	1	no	—
530	C₂²	—	268	1	no	—
531	C₂²	—	268	1	no	—
532	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
533	C₂²	#16864109	1	1	yes	characteristic
534	C₂²	#16864114	1	1	yes	characteristic
535	C₄	—	2	1	no	—
536	C₄	—	2	1	no	—
537	C₄	—	2	1	no	—
538	C₄	—	4	1	no	—
539	C₄	—	4	1	no	—
540	C₄	—	4	1	no	—
541	C₄	—	268	1	no	—
542	C₄	—	268	1	no	—
543	C₄	—	268	1	no	—
544	C₄	—	536	2	no	—
545	C₄	#16863516	1	2	yes	—
546	C₄	#117×D₆₇	1	1	yes	characteristic
547	C₄	#16863876	1	1	yes	characteristic
548	C₄	#16864045	1	1	yes	characteristic
549	C₄	#16864065	1	1	yes	characteristic
550	C₂	—	2	1	no	—
551	C₂	—	2	1	no	—
552	C₂	—	8	1	no	—
553	C₂	—	536	1	no	—
554	C₂	#22563159	1	1	yes	characteristic
555	C₂	C₂×#16863516	1	1	yes	characteristic
556	C₂	#22569124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2925	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2925	D₆₇
hypocenter	C₆₇	#22803
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #2925

Complements

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

N	H ≅ G/N	type
2. #22568874	C₂	semidirect
4. #22560857	C₂	semidirect
5. #22560854	C₂	semidirect
6. #22560380	C₂	semidirect
7. #22560229	C₂	semidirect
8. #22560226	C₂	semidirect
9. #22560094	C₂	semidirect
10. #22560079	C₂	semidirect
11. C₆₇×#2925	C₂	semidirect
14. C₄×#11761618	C₄	semidirect
15. C₄×#11761618	C₄	semidirect
16. C₄×#11761615	C₄	semidirect
17. C₄×#11761615	C₄	semidirect
18. C₄×#11761616	C₄	semidirect
19. C₄×#11761616	C₄	semidirect
20. C₄×#11761613	C₄	semidirect
21. C₄×#11761613	C₄	semidirect
281. #2925	D₆₇	semidirect
290. C₆₇	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524063	2
#22568874	2
#22561101	2
#22560857	2
#22560854	2
#22560380	2
#22560229	2
#22560226	2
#22560094	2
#22560079	2
C₆₇×#2925	2
#22803	67