Subgroups — #30877907

Order 17152.

708 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569997	C₂	1	1	yes	maximal, characteristic
2	#22569156	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568777	C₂	1	1	yes	maximal, characteristic
4	#22561186	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560984	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560961	C₂	1	2	yes	maximal, complemented
7	#22560811	C₂	1	2	yes	maximal, complemented
8	#22560482	C₂	1	2	yes	maximal
9	#22560062	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560054	C₂	1	2	yes	maximal
11	C₄×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₂²	1	2	yes	—
13	#16863953	C₂²	1	1	yes	characteristic
14	#16863703	C₄	1	2	yes	complemented
15	#16863696	C₄	1	2	yes	complemented
16	#16863691	C₄	1	1	yes	characteristic, complemented
17	#16863686	C₄	1	1	yes	characteristic, complemented
18	#16863681	C₄	1	1	yes	characteristic, complemented
19	#16863676	C₄	1	1	yes	characteristic, complemented
20	#16863526	—	2	1	no	—
21	#16863524	—	2	2	no	—
22	C₈×Dic₁₃₄	—	2	1	no	—
23	C₂×#14523978	C₂²	1	2	yes	—
24	#16863432	C₂²	1	2	yes	—
25	#16863424	C₂²	1	2	yes	—
26	#16863423	—	2	2	no	—
27	#16863423	C₂²	1	2	yes	—
28	#16863420	—	2	1	no	—
29	#16863420	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
31	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
33	#16863392	C₂²	1	2	yes	—
34	#16863354	C₂²	1	2	yes	—
35	C₂×#14523954	C₂²	1	2	yes	—
36	#16863160	C₂²	1	2	yes	—
37	#16863156	C₂²	1	2	yes	—
38	#16863156	C₂²	1	2	yes	—
39	#16863151	C₂²	1	2	yes	—
40	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#104×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
44	C₄³×C₆₇	C₂²	1	1	yes	characteristic
45	#339×C₆₇	C₂²	1	1	yes	characteristic
46	#339×C₆₇	C₂²	1	1	yes	characteristic
47	#14524050	—	2	4	no	—
48	#14524050	—	4	2	no	—
49	C₂×#11761612	—	2	2	no	—
50	C₂×#11761612	C₂³	1	2	yes	—
51	C₂²×C₆₇⋊C₈	—	2	2	no	—
52	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
54	#14524043	C₂×C₄	1	1	yes	characteristic
55	#14524043	C₂×C₄	1	1	yes	characteristic
56	#14524035	C₂×C₄	1	2	yes	—
57	#14524022	—	2	1	no	—
58	#14524022	—	2	1	no	—
59	C₄×Dic₁₃₄	—	2	1	no	—
60	C₄×Dic₁₃₄	—	2	1	no	—
61	#14524002	C₂³	1	2	yes	—
62	#14523978	—	4	2	no	—
63	#14523976	C₂×C₄	1	2	yes	—
64	#14523976	C₂×C₄	1	2	yes	—
65	#14523974	C₂×C₄	1	2	yes	—
66	#14523974	C₂×C₄	1	2	yes	—
67	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
68	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
69	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
70	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
71	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
72	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
73	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
74	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
75	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
76	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	2	no	—
79	#14523967	C₂×C₄	1	2	yes	—
80	#14523967	C₂×C₄	1	2	yes	—
81	#14523964	C₂×C₄	1	2	yes	—
82	#14523964	C₂×C₄	1	2	yes	—
83	#14523954	—	4	2	no	—
84	#14523953	—	2	2	no	—
85	C₈×Dic₆₇	—	2	2	no	—
86	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₄×D₄×C₆₇	—	2	1	no	—
92	C₄×D₄×C₆₇	—	2	1	no	—
93	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
95	#117×C₆₇	—	2	1	no	—
96	#117×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	—	2	1	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	#104×C₆₇	—	2	2	no	—
104	#104×C₆₇	Q₈	1	4	yes	complemented
105	#97×C₆₇	—	2	1	no	—
106	C₄×C₈×C₆₇	—	2	1	no	—
107	C₂×Dic₁₃₄	—	2	1	no	—
108	C₂×Dic₁₃₄	—	2	1	no	—
109	#11761612	—	2	2	no	—
110	#11761612	—	2	2	no	—
111	#11761612	—	4	4	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	2	no	—
115	C₂×C₆₇⋊C₈	—	2	2	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	—	4	4	no	—
118	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
120	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
121	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
122	C₄⋊Dic₆₇	—	2	1	no	—
123	C₄⋊Dic₆₇	—	2	1	no	—
124	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
125	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
126	Dic₆₇⋊C₄	—	2	2	no	—
127	Dic₆₇⋊C₄	—	2	2	no	—
128	C₂²⋊Dic₆₇	—	2	2	no	—
129	C₄×Dic₆₇	—	2	2	no	—
130	#41×C₆₇	—	2	1	no	—
131	#41×C₆₇	—	2	1	no	—
132	#41×C₆₇	C₂×Q₈	1	2	yes	—
133	#41×C₆₇	#41	1	2	yes	—
134	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂×D₄×C₆₇	—	2	1	no	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	M₁₆×C₆₇	—	2	1	no	—
146	M₁₆×C₆₇	—	2	1	no	—
147	M₁₆×C₆₇	—	2	1	no	—
148	M₁₆×C₆₇	C₂×Q₈	1	2	yes	—
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	—	2	1	no	—
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂²⋊C₄×C₆₇	—	2	1	no	—
156	C₂²⋊C₄×C₆₇	—	2	1	no	—
157	C₂²⋊C₄×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	—	2	1	no	—
162	C₄²×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	4	2	no	—
165	C₄²×C₆₇	—	4	2	no	—
166	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
173	Q₈×C₆₇	—	2	1	no	—
174	Q₈×C₆₇	C₄×Q₈	1	2	yes	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	—	2	2	no	—
177	D₄×C₆₇	—	2	1	no	—
178	D₄×C₆₇	C₄×Q₈	1	2	yes	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	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	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	8	1	no	—
194	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
204	C₂×Dic₆₇	—	2	2	no	—
205	C₂×Dic₆₇	—	2	2	no	—
206	Dic₁₃₄	—	4	2	no	—
207	Dic₁₃₄	—	8	1	no	—
208	C₈×C₆₇	—	2	2	no	—
209	C₈×C₆₇	—	4	1	no	—
210	C₆₇⋊C₈	—	2	2	no	—
211	C₆₇⋊C₈	—	2	2	no	—
212	C₆₇⋊C₈	—	2	2	no	—
213	C₆₇⋊C₈	—	2	2	no	—
214	C₆₇⋊C₈	—	4	2	no	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₆₇	#552	1	1	yes	characteristic
222	C₂²×C₆₇	#554	1	1	yes	characteristic
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	—	4	2	no	—
229	C₄×C₆₇	—	4	1	no	—
230	C₄×C₆₇	—	4	1	no	—
231	C₄×C₆₇	#444	1	2	yes	—
232	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₄×C₆₇	#520	1	1	yes	characteristic
234	C₄×C₆₇	#546	1	1	yes	characteristic
235	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
236	Dic₆₇	—	4	2	no	—
237	Dic₆₇	—	4	2	no	—
238	#22983	—	67	1	no	maximal, Sylow
239	C₂×C₆₇	—	2	1	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	—	4	2	no	—
242	C₂×C₆₇	#2360	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
244	C₂×C₆₇	#2956	1	1	yes	characteristic
245	#3076	—	67	1	no	—
246	#2962	—	67	1	no	—
247	#2906	—	67	1	no	—
248	#1994	—	67	1	no	—
249	#1950	—	67	1	no	—
250	#1946	—	67	2	no	—
251	#1922	—	67	2	no	—
252	#1866	—	67	2	no	—
253	#1767	—	67	1	no	—
254	#1765	—	67	2	no	—
255	C₄×#104	D₆₇	1	1	yes	characteristic
256	C₆₇	#22983	1	1	yes	characteristic, complemented, Sylow
257	C₂×#131	—	67	2	no	—
258	#533	—	67	1	no	—
259	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
260	#479	—	67	2	no	—
261	#478	—	67	2	no	—
262	#477	—	67	1	no	—
263	#476	—	67	1	no	—
264	#475	—	67	1	no	—
265	#474	—	67	1	no	—
266	#446	—	134	1	no	—
267	#446	—	134	2	no	—
268	C₈×Q₈	—	134	1	no	—
269	C₂×#108	—	67	2	no	—
270	#427	—	67	2	no	—
271	#424	—	67	2	no	—
272	#424	—	67	2	no	—
273	#424	—	134	2	no	—
274	#423	—	67	1	no	—
275	#423	—	134	1	no	—
276	C₂×C₄⋊C₈	—	67	1	no	—
277	C₂×C₄⋊C₈	—	67	1	no	—
278	C₂×C₄⋊C₈	—	134	1	no	—
279	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
280	C₂×#104	C₂×D₆₇	1	1	yes	characteristic
281	#416	—	67	2	no	—
282	#405	—	67	2	no	—
283	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
284	C₂×#97	—	67	2	no	—
285	C₄³	C₂×D₆₇	1	1	yes	characteristic
286	#341	—	67	2	no	—
287	#339	—	67	2	no	—
288	#339	—	67	2	no	—
289	#339	C₂×D₆₇	1	1	yes	characteristic
290	#339	C₂×D₆₇	1	1	yes	characteristic
291	#336	—	67	2	no	—
292	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
293	#131	—	268	2	no	—
294	#131	—	134	4	no	—
295	C₂×M₁₆	—	67	2	no	—
296	C₂×M₁₆	—	134	2	no	—
297	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
298	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
299	C₂²×C₈	—	67	2	no	—
300	C₂²×C₈	—	67	2	no	—
301	C₂²×C₈	—	134	2	no	—
302	C₄⋊Q₈	—	67	1	no	—
303	C₄⋊Q₈	—	67	1	no	—
304	#125	—	67	2	no	—
305	C₂²⋊Q₈	—	134	1	no	—
306	C₂²⋊Q₈	—	134	1	no	—
307	C₄×Q₈	—	134	1	no	—
308	C₄×Q₈	—	134	1	no	—
309	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
310	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
314	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
315	#117	—	2	1	no	—
316	#117	—	67	2	no	—
317	#117	C₂²×D₆₇	1	1	yes	characteristic
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
323	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
324	#108	—	268	2	no	—
325	#107	—	67	2	no	—
326	#107	—	67	2	no	—
327	#106	—	67	2	no	—
328	#106	—	67	2	no	—
329	C₄⋊C₈	—	67	1	no	—
330	C₄⋊C₈	—	67	1	no	—
331	C₄⋊C₈	—	67	2	no	—
332	C₄⋊C₈	—	67	1	no	—
333	C₄⋊C₈	—	67	1	no	—
334	C₄⋊C₈	—	67	2	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	C₄⋊C₈	—	134	2	no	—
339	C₄⋊C₈	—	134	1	no	—
340	C₄⋊C₈	—	134	1	no	—
341	#104	—	2	2	no	—
342	#104	Dic₁₃₄	1	4	yes	complemented
343	#103	—	67	2	no	—
344	#103	—	67	2	no	—
345	#102	—	67	2	no	—
346	#102	—	67	2	no	—
347	#97	—	2	1	no	—
348	#97	—	268	2	no	—
349	#97	—	134	2	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	—	134	2	no	—
352	#41	—	2	1	no	—
353	#41	—	2	1	no	—
354	#41	C₂×Dic₁₃₄	1	2	yes	—
355	#41	#11761628	1	2	yes	—
356	C₂×Q₈	—	134	1	no	—
357	C₂×Q₈	—	134	1	no	—
358	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	#11761627	1	1	yes	characteristic
367	C₂²×C₄	#11761627	1	1	yes	characteristic
368	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
369	M₁₆	—	2	1	no	—
370	M₁₆	—	2	1	no	—
371	M₁₆	—	2	1	no	—
372	M₁₆	—	268	4	no	—
373	M₁₆	—	134	2	no	—
374	M₁₆	—	134	2	no	—
375	M₁₆	C₂×Dic₁₃₄	1	2	yes	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	268	4	no	—
379	C₂×C₈	—	67	2	no	—
380	C₂×C₈	—	67	2	no	—
381	C₂×C₈	—	67	2	no	—
382	C₂×C₈	—	67	2	no	—
383	C₂×C₈	—	134	2	no	—
384	C₂×C₈	—	134	2	no	—
385	C₂×C₈	—	134	2	no	—
386	C₂×C₈	—	134	2	no	—
387	C₂×C₈	—	134	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	67	2	no	—
391	C₄⋊C₄	—	67	2	no	—
392	C₄⋊C₄	—	134	2	no	—
393	C₄⋊C₄	—	134	2	no	—
394	C₄⋊C₄	—	134	1	no	—
395	C₄⋊C₄	—	134	1	no	—
396	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
397	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	1	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	134	2	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	4	2	no	—
409	C₄²	—	4	2	no	—
410	C₄²	—	134	2	no	—
411	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
412	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
413	C₄²	D₄×D₆₇	1	1	yes	characteristic
414	C₄²	D₄×D₆₇	1	1	yes	characteristic
415	C₄²	C₂×Dic₁₃₄	1	2	yes	—
416	C₂³	—	2	1	no	—
417	C₂³	C₂×#11761627	1	1	yes	characteristic
418	Q₈	—	2	1	no	—
419	Q₈	—	268	2	no	—
420	Q₈	—	536	1	no	—
421	Q₈	C₄×Dic₁₃₄	1	2	yes	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	1	no	—
425	D₄	C₄×Dic₁₃₄	1	2	yes	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	8	1	no	—
441	C₂×C₄	—	134	2	no	—
442	C₂×C₄	—	134	2	no	—
443	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
444	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
445	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
446	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
447	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
450	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
451	C₂×C₄	#14524089	1	1	yes	characteristic
452	C₂×C₄	#14524090	1	1	yes	characteristic
453	C₈	—	2	2	no	—
454	C₈	—	4	1	no	—
455	C₈	—	268	2	no	—
456	C₈	—	134	2	no	—
457	C₈	—	134	2	no	—
458	C₈	—	134	2	no	—
459	C₈	—	134	2	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
466	C₂²	#16864086	1	1	yes	characteristic
467	C₂²	#16864103	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	268	2	no	—
477	C₄	—	268	2	no	—
478	C₄	#16863515	1	2	yes	—
479	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
480	C₄	#16863876	1	1	yes	characteristic
481	C₄	#16864042	1	1	yes	characteristic
482	C₄	D₄×Dic₁₃₄	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	4	2	no	—
486	C₂	#22563212	1	1	yes	characteristic
487	C₂	C₂×#16863515	1	1	yes	characteristic
488	C₂	#22569115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₄×#104×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×#104	D₆₇
hypocenter	C₆₇	#22983
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#104

Complements

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

N	H ≅ G/N	type
2. #22569156	C₂	semidirect
4. #22561186	C₂	semidirect
5. #22560984	C₂	semidirect
6. #22560961	C₂	semidirect
7. #22560811	C₂	semidirect
9. #22560062	C₂	semidirect
14. #16863703	C₄	semidirect
15. #16863696	C₄	semidirect
16. #16863691	C₄	semidirect
17. #16863686	C₄	semidirect
18. #16863681	C₄	semidirect
19. #16863676	C₄	semidirect
104. #104×C₆₇	Q₈	semidirect
256. C₆₇	#22983	semidirect
342. #104	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22569997	2
#22569156	2
#22568777	2
#22561186	2
#22560984	2
#22560961	2
#22560811	2
#22560482	2
#22560062	2
#22560054	2
C₄×#104×C₆₇	2
#22983	67