Subgroups — #30874358

Order 17152.

808 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569124	C₂	1	2	yes	maximal
2	C₂×#16863516	C₂	1	1	yes	maximal, characteristic
3	#22568817	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560996	C₂	1	2	yes	maximal, complemented
5	#22560449	C₂	1	2	yes	maximal, complemented
6	#22560446	C₂	1	2	yes	maximal, complemented
7	#22560382	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560264	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560263	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560096	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	#16864162	C₂²	1	2	yes	—
14	C₂×#14524049	C₂²	1	1	yes	characteristic
15	#16863516	—	2	4	no	—
16	#16863516	C₄	1	4	yes	complemented
17	#16863516	C₄	1	4	yes	complemented
18	#16863511	—	2	4	no	—
19	#16863444	C₂²	1	2	yes	—
20	#16863443	—	2	2	no	—
21	C₂×#14523979	C₂²	1	1	yes	characteristic
22	C₂×#14523978	C₂²	1	2	yes	—
23	C₂×#14523978	C₂²	1	1	yes	characteristic
24	#108×D₆₇	—	2	2	no	—
25	#16863434	C₂²	1	1	yes	characteristic
26	#16863419	C₂²	1	2	yes	—
27	#16863413	C₂²	1	2	yes	—
28	#16863412	C₂²	1	2	yes	—
29	C₂×#14523970	C₂²	1	1	yes	characteristic
30	C₂×#14523969	C₂²	1	1	yes	characteristic
31	#16863354	C₂²	1	2	yes	—
32	C₂×#14523953	C₂²	1	1	yes	characteristic
33	#16863168	C₂²	1	2	yes	—
34	#16863167	C₂²	1	2	yes	—
35	#16863157	C₂²	1	2	yes	—
36	#568×C₆₇	C₂²	1	2	yes	—
37	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
38	#430×C₆₇	—	2	2	no	—
39	#430×C₆₇	C₂²	1	2	yes	—
40	C₂×#108×C₆₇	—	2	2	no	—
41	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#11761628	C₂³	1	1	yes	characteristic
44	#14524063	C₂×C₄	1	4	yes	—
45	#14524063	D₄	1	4	yes	—
46	C₂×#11761618	Q₈	1	1	yes	characteristic
47	C₂×#11761616	Q₈	1	1	yes	characteristic
48	C₂×#11761615	Q₈	1	1	yes	characteristic
49	C₂×#11761613	Q₈	1	1	yes	characteristic
50	#14524051	—	4	2	no	—
51	#14524049	—	2	4	no	—
52	#14524049	—	2	1	no	—
53	#14524049	—	2	1	no	—
54	#14524049	C₂×C₄	1	2	yes	—
55	#14524049	C₂×C₄	1	2	yes	—
56	C₂×#11761612	C₂³	1	2	yes	—
57	C₂×#11761612	C₂³	1	1	yes	characteristic
58	C₂×#11761611	—	2	1	no	—
59	M₁₆×D₆₇	—	2	4	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₆₇	—	2	1	no	—
62	#14524002	C₂³	1	2	yes	—
63	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
64	#14523979	—	2	1	no	—
65	#14523979	—	2	1	no	—
66	#14523978	—	2	2	no	—
67	#14523978	—	2	2	no	—
68	#14523978	—	2	4	no	—
69	#14523978	C₂×C₄	1	4	yes	—
70	#14523977	—	2	1	no	—
71	#14523975	—	2	1	no	—
72	Dic₆₇⋊C₈	—	4	2	no	—
73	#14523970	—	2	4	no	—
74	#14523970	C₂×C₄	1	4	yes	—
75	#14523969	—	2	4	no	—
76	#14523969	C₂×C₄	1	4	yes	—
77	#14523953	—	2	2	no	—
78	#14523953	C₂×C₄	1	2	yes	—
79	#14523953	C₂×C₄	1	2	yes	—
80	C₈×Dic₆₇	—	2	2	no	—
81	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
82	#131×C₆₇	—	2	1	no	—
83	#131×C₆₇	—	2	1	no	—
84	#131×C₆₇	—	2	4	no	—
85	#131×C₆₇	—	4	2	no	—
86	#131×C₆₇	C₂×C₄	1	2	yes	—
87	#131×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₆₇	—	2	1	no	—
89	C₂×M₁₆×C₆₇	—	2	4	no	—
90	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
93	C₂²×C₈×C₆₇	—	2	1	no	—
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#108×C₆₇	—	2	4	no	—
96	#108×C₆₇	—	2	1	no	—
97	#108×C₆₇	—	2	1	no	—
98	#108×C₆₇	D₄	1	2	yes	—
99	#108×C₆₇	D₄	1	2	yes	—
100	#11761628	—	2	1	no	—
101	#11761628	—	2	1	no	—
102	#11761628	C₂²×C₄	1	2	yes	—
103	#11761628	C₂×D₄	1	2	yes	—
104	C₂×Dic₁₃₄	C₂×Q₈	1	1	yes	characteristic
105	C₂×D₂₆₈	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₂²⋊D₆₇	—	2	1	no	—
107	C₂²×Dic₆₇	—	2	1	no	—
108	C₂×C₄×D₆₇	—	2	1	no	—
109	#11761618	C₄⋊C₄	1	4	yes	—
110	#11761616	C₄⋊C₄	1	4	yes	—
111	#11761615	C₄⋊C₄	1	4	yes	—
112	#11761613	C₄⋊C₄	1	4	yes	—
113	#11761612	—	2	1	no	—
114	#11761612	—	2	2	no	—
115	#11761612	—	2	2	no	—
116	#11761612	C₂²×C₄	1	2	yes	—
117	#11761611	—	2	2	no	—
118	#11761611	—	2	2	no	—
119	#11761611	—	4	4	no	—
120	C₂×C₆₇⋊C₈	—	2	1	no	—
121	C₂×C₆₇⋊C₈	—	2	2	no	—
122	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
124	C₂×C₆₇⋊C₈	C₂×Q₈	1	1	yes	characteristic
125	C₂×C₆₇⋊C₈	C₂×Q₈	1	1	yes	characteristic
126	C₈×D₆₇	—	2	2	no	—
127	C₈×D₆₇	—	2	2	no	—
128	C₈×D₆₇	—	4	4	no	—
129	C₄⋊Dic₆₇	—	2	2	no	—
130	C₂²⋊Dic₆₇	—	2	2	no	—
131	C₄×Dic₆₇	—	2	1	no	—
132	C₄×Dic₆₇	—	2	2	no	—
133	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
134	#41×C₆₇	—	2	1	no	—
135	#41×C₆₇	—	2	1	no	—
136	#41×C₆₇	C₂²×C₄	1	2	yes	—
137	#41×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
139	C₂×D₄×C₆₇	—	2	1	no	—
140	C₂×D₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	M₁₆×C₆₇	—	2	1	no	—
144	M₁₆×C₆₇	—	2	2	no	—
145	M₁₆×C₆₇	—	2	2	no	—
146	M₁₆×C₆₇	—	2	2	no	—
147	M₁₆×C₆₇	—	2	2	no	—
148	M₁₆×C₆₇	—	2	2	no	—
149	M₁₆×C₆₇	—	4	4	no	—
150	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
151	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	4	4	no	—
158	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
163	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
164	C₂³×C₆₇	—	2	1	no	—
165	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂²×D₆₇	—	2	1	no	—
167	Q₈×C₆₇	—	2	1	no	—
168	Q₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
169	D₄×C₆₇	—	2	2	no	—
170	D₄×C₆₇	—	2	2	no	—
171	D₄×C₆₇	—	2	1	no	—
172	D₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂²⋊D₆₇	—	2	2	no	—
183	C₂²⋊D₆₇	—	2	2	no	—
184	C₂×Dic₆₇	—	2	1	no	—
185	C₂×Dic₆₇	—	2	2	no	—
186	C₂×Dic₆₇	—	4	2	no	—
187	C₂×Dic₆₇	—	8	1	no	—
188	D₂₆₈	—	2	1	no	—
189	D₂₆₈	C₂×C₄⋊C₄	1	2	yes	—
190	C₄×D₆₇	—	2	2	no	—
191	C₄×D₆₇	—	2	2	no	—
192	Dic₁₃₄	—	2	1	no	—
193	Dic₁₃₄	C₂×C₄⋊C₄	1	2	yes	—
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	—	2	4	no	—
196	C₈×C₆₇	—	4	2	no	—
197	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₆₇⋊C₈	—	2	2	no	—
202	C₆₇⋊C₈	—	4	2	no	—
203	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
204	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
205	C₂²×C₆₇	—	2	2	no	—
206	C₂²×C₆₇	—	4	1	no	—
207	C₂²×C₆₇	—	8	1	no	—
208	C₂²×C₆₇	#520	1	1	yes	characteristic
209	C₂²×C₆₇	#547	1	1	yes	characteristic
210	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
211	C₂×D₆₇	—	2	2	no	—
212	C₂×D₆₇	—	8	1	no	—
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
215	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₆₇	D₄²	1	1	yes	characteristic
217	C₄×C₆₇	#554	1	1	yes	characteristic
218	Dic₆₇	—	2	2	no	—
219	Dic₆₇	—	4	2	no	—
220	Dic₆₇	—	8	2	no	—
221	#22695	—	67	1	no	maximal, Sylow
222	C₂×C₆₇	—	2	1	no	—
223	C₂×C₆₇	—	2	1	no	—
224	C₂×C₆₇	—	2	1	no	—
225	C₂×C₆₇	—	4	2	no	—
226	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
227	D₆₇	—	4	2	no	—
228	#2957	—	67	2	no	—
229	C₂×#444	—	67	1	no	—
230	#2916	—	67	1	no	—
231	#2916	D₆₇	1	1	yes	characteristic, complemented
232	#1953	—	67	2	no	—
233	#1861	—	67	2	no	—
234	#1861	—	67	2	no	—
235	#1849	—	67	1	no	—
236	#1814	—	67	1	no	—
237	#1814	—	67	1	no	—
238	#1773	—	67	1	no	—
239	C₆₇	#22695	1	1	yes	characteristic, complemented, Sylow
240	C₂×#135	—	67	1	no	—
241	#568	—	67	2	no	—
242	#568	C₂×D₆₇	1	2	yes	—
243	C₂×#131	—	67	1	no	—
244	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
245	#444	—	67	4	no	—
246	#444	—	67	4	no	—
247	#444	—	134	4	no	—
248	#443	—	134	4	no	—
249	#430	—	2	2	no	—
250	#430	—	67	2	no	—
251	#430	—	134	2	no	—
252	#430	C₂×D₆₇	1	2	yes	—
253	C₂×#108	—	2	2	no	—
254	C₂×#108	—	67	1	no	—
255	C₂×#108	—	67	2	no	—
256	C₂×#108	—	67	1	no	—
257	C₂×#108	—	134	2	no	—
258	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
259	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
260	#427	—	67	1	no	—
261	#423	—	67	2	no	—
262	#421	—	67	2	no	—
263	#421	—	67	2	no	—
264	C₂×#104	—	67	1	no	—
265	C₂×#104	—	67	1	no	—
266	#405	—	67	2	no	—
267	C₂×#97	—	67	1	no	—
268	#344	—	67	2	no	—
269	#344	—	67	2	no	—
270	#339	—	67	2	no	—
271	C₂×#41	—	67	1	no	—
272	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
273	#135	—	67	4	no	—
274	#135	—	67	4	no	—
275	C₂×Q₁₆	—	67	1	no	—
276	C₂×SD₁₆	—	67	1	no	—
277	C₂×SD₁₆	—	67	1	no	—
278	C₂×D₈	—	67	1	no	—
279	#131	—	2	1	no	—
280	#131	—	2	1	no	—
281	#131	—	2	4	no	—
282	#131	—	4	2	no	—
283	#131	—	268	2	no	—
284	#131	—	67	2	no	—
285	#131	—	67	2	no	—
286	#131	—	134	4	no	—
287	#131	—	134	1	no	—
288	#131	—	134	1	no	—
289	#131	C₄×D₆₇	1	2	yes	—
290	#131	C₄×D₆₇	1	2	yes	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	—	2	4	no	—
293	C₂×M₁₆	—	67	1	no	—
294	C₂×M₁₆	—	67	2	no	—
295	C₂×M₁₆	—	134	4	no	—
296	C₂×M₁₆	—	134	1	no	—
297	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
298	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
299	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
300	C₂²×C₈	—	2	1	no	—
301	C₂²×C₈	—	67	1	no	—
302	C₂²×C₈	—	134	1	no	—
303	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
304	#117	—	67	2	no	—
305	C₂×C₄²	—	67	1	no	—
306	#108	—	2	4	no	—
307	#108	—	2	1	no	—
308	#108	—	2	1	no	—
309	#108	—	67	4	no	—
310	#108	—	134	2	no	—
311	#108	—	134	2	no	—
312	#108	—	134	4	no	—
313	#108	—	134	1	no	—
314	#108	—	134	1	no	—
315	#108	C₂²⋊D₆₇	1	2	yes	—
316	#108	C₂²⋊D₆₇	1	2	yes	—
317	#107	—	134	1	no	—
318	#106	—	134	1	no	—
319	C₄⋊C₈	—	268	2	no	—
320	#104	—	67	4	no	—
321	#104	—	67	4	no	—
322	#104	—	134	4	no	—
323	#104	—	134	4	no	—
324	#97	—	67	2	no	—
325	#97	—	67	2	no	—
326	#97	—	134	2	no	—
327	C₄×C₈	—	134	2	no	—
328	#41	—	2	1	no	—
329	#41	—	2	1	no	—
330	#41	—	67	2	no	—
331	#41	—	67	2	no	—
332	#41	—	134	1	no	—
333	#41	—	134	1	no	—
334	#41	C₂×C₄×D₆₇	1	2	yes	—
335	#41	D₄×D₆₇	1	2	yes	—
336	C₂×Q₈	—	67	1	no	—
337	C₂×Q₈	Q₈×D₆₇	1	1	yes	characteristic
338	C₂×D₄	—	2	1	no	—
339	C₂×D₄	—	67	1	no	—
340	C₂×D₄	—	134	1	no	—
341	C₂×D₄	Q₈×D₆₇	1	1	yes	characteristic
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	134	1	no	—
344	C₂²×C₄	—	134	1	no	—
345	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
346	Q₁₆	—	67	4	no	—
347	SD₁₆	—	67	4	no	—
348	SD₁₆	—	67	4	no	—
349	D₈	—	67	4	no	—
350	M₁₆	—	2	1	no	—
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	2	2	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	4	4	no	—
357	M₁₆	—	268	4	no	—
358	M₁₆	—	67	2	no	—
359	M₁₆	—	134	2	no	—
360	M₁₆	—	134	2	no	—
361	M₁₆	—	134	1	no	—
362	M₁₆	—	134	2	no	—
363	M₁₆	—	134	2	no	—
364	M₁₆	C₂×C₄×D₆₇	1	2	yes	—
365	M₁₆	C₂×C₂²⋊D₆₇	1	4	yes	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	4	no	—
372	C₂×C₈	—	268	4	no	—
373	C₂×C₈	—	67	2	no	—
374	C₂×C₈	—	67	2	no	—
375	C₂×C₈	—	67	1	no	—
376	C₂×C₈	—	67	1	no	—
377	C₂×C₈	—	134	2	no	—
378	C₂×C₈	—	134	1	no	—
379	C₂×C₈	—	134	2	no	—
380	C₂×C₈	—	134	2	no	—
381	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
382	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
383	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
384	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
385	C₂×C₈	#11761628	1	1	yes	characteristic
386	C₂×C₈	#11761628	1	1	yes	characteristic
387	C₄⋊C₄	—	134	2	no	—
388	C₂²⋊C₄	—	134	2	no	—
389	C₄²	—	67	2	no	—
390	C₄²	—	134	2	no	—
391	C₄²	—	134	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	134	1	no	—
394	C₂³	#14524091	1	1	yes	characteristic
395	Q₈	—	2	1	no	—
396	Q₈	—	67	2	no	—
397	Q₈	—	134	1	no	—
398	Q₈	C₄⋊C₄×D₆₇	1	2	yes	—
399	D₄	—	2	2	no	—
400	D₄	—	2	2	no	—
401	D₄	—	2	1	no	—
402	D₄	—	67	2	no	—
403	D₄	—	134	2	no	—
404	D₄	—	134	2	no	—
405	D₄	—	134	1	no	—
406	D₄	C₄⋊C₄×D₆₇	1	2	yes	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	268	2	no	—
411	C₂×C₄	—	536	1	no	—
412	C₂×C₄	—	134	2	no	—
413	C₂×C₄	—	134	1	no	—
414	C₂×C₄	—	134	2	no	—
415	C₂×C₄	—	134	2	no	—
416	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
417	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
418	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
420	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
421	C₂×C₄	#14524088	1	1	yes	characteristic
422	C₈	—	2	4	no	—
423	C₈	—	2	2	no	—
424	C₈	—	4	2	no	—
425	C₈	—	268	2	no	—
426	C₈	—	67	2	no	—
427	C₈	—	67	2	no	—
428	C₈	—	134	2	no	—
429	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
430	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
431	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
432	C₈	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
433	C₂²	—	2	2	no	—
434	C₂²	—	4	1	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	536	1	no	—
437	C₂²	—	134	2	no	—
438	C₂²	#16863877	1	1	yes	characteristic
439	C₂²	#16864047	1	1	yes	characteristic
440	C₂²	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
441	C₄	—	2	2	no	—
442	C₄	—	268	2	no	—
443	C₄	—	536	2	no	—
444	C₄	—	134	2	no	—
445	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
446	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
447	C₄	D₄×C₂²⋊D₆₇	1	1	yes	characteristic
448	C₄	#16864101	1	1	yes	characteristic
449	C₂	—	2	1	no	—
450	C₂	—	2	1	no	—
451	C₂	—	2	1	no	—
452	C₂	—	4	2	no	—
453	C₂	—	268	2	no	—
454	C₂	C₄⋊C₄×C₂²⋊D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#2916	C₂
socle	C₂×C₆₇	D₄×C₄⋊C₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2916	D₆₇
hypocenter	C₆₇	#22695
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2916

Complements

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

N	H ≅ G/N	type
3. #22568817	C₂	semidirect
4. #22560996	C₂	semidirect
5. #22560449	C₂	semidirect
6. #22560446	C₂	semidirect
7. #22560382	C₂	semidirect
8. #22560264	C₂	semidirect
9. #22560263	C₂	semidirect
10. #22560096	C₂	semidirect
11. C₆₇×#2916	C₂	semidirect
16. #16863516	C₄	semidirect
17. #16863516	C₄	semidirect
231. #2916	D₆₇	semidirect
239. C₆₇	#22695	semidirect

Maximal subgroups

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

subgroup	index
#22569124	2
C₂×#16863516	2
#22568817	2
#22560996	2
#22560449	2
#22560446	2
#22560382	2
#22560264	2
#22560263	2
#22560096	2
C₆₇×#2916	2
#22695	67