Subgroups — #30825834

Order 17152.

632 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570281	C₂	1	1	yes	maximal, characteristic
2	#22569165	C₂	1	1	yes	maximal, characteristic
3	#22569139	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559079	C₂	1	2	yes	maximal, complemented
5	#22559065	C₂	1	2	yes	maximal, complemented
6	#22559015	C₂	1	2	yes	maximal, complemented
7	#22559008	C₂	1	2	yes	maximal, complemented
8	#22558646	C₂	1	1	yes	maximal, characteristic
9	#22558629	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863953	C₂²	1	1	yes	characteristic
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863738	C₄	1	2	yes	—
15	#16863682	C₄	1	1	yes	characteristic
16	#16863677	C₄	1	1	yes	characteristic
17	#16863607	C₄	1	1	yes	characteristic
18	#16863602	C₄	1	1	yes	characteristic
19	#16863560	C₄	1	2	yes	—
20	#16863523	—	2	1	no	—
21	#16863523	C₂²	1	2	yes	—
22	#16863520	—	2	1	no	—
23	#16863520	C₂²	1	2	yes	—
24	#16863520	C₂²	1	1	yes	characteristic
25	#16863520	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₆₈	C₂²	1	2	yes	—
27	C₄×#11761617	C₂²	1	2	yes	—
28	#16863461	C₂²	1	1	yes	characteristic
29	#16863461	C₂²	1	1	yes	characteristic
30	(C₂²⋊D₆₇)⋊C₈	—	2	1	no	—
31	(C₂²⋊D₆₇)⋊C₈	—	2	1	no	—
32	#16863447	—	2	1	no	—
33	#16863424	—	2	1	no	—
34	#16863354	—	2	1	no	—
35	C₄×#11761612	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
37	C₄×C₈×D₆₇	—	2	1	no	—
38	#16863126	C₂²	1	2	yes	—
39	#16863126	C₂²	1	2	yes	—
40	#16863126	C₂²	1	2	yes	—
41	#16863123	C₂²	1	2	yes	—
42	#16863118	C₂²	1	2	yes	—
43	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
44	#431×C₆₇	C₂²	1	1	yes	characteristic
45	#431×C₆₇	C₂²	1	1	yes	characteristic
46	#423×C₆₇	C₂²	1	1	yes	characteristic
47	#335×C₆₇	C₂²	1	2	yes	—
48	#334×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
50	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
51	C₂×#11761617	C₂×C₄	1	2	yes	—
52	C₂×#11761612	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
54	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
55	C₂×C₈×D₆₇	—	2	1	no	—
56	C₂×C₈×D₆₇	—	2	1	no	—
57	#14524043	C₂×C₄	1	1	yes	characteristic
58	#14524043	C₂×C₄	1	1	yes	characteristic
59	#14524035	—	2	1	no	—
60	#14524022	—	2	1	no	—
61	#14524022	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃₄	—	2	1	no	—
63	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₃₄	C₂³	1	2	yes	—
66	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₆₈	C₂³	1	1	yes	characteristic
68	C₄×C₂²⋊D₆₇	—	2	1	no	—
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	#14524002	—	2	1	no	—
71	#14524002	C₂×C₄	1	1	yes	characteristic
72	#14524000	—	2	1	no	—
73	C₄²×D₆₇	—	2	1	no	—
74	#14523977	C₂×C₄	1	2	yes	—
75	#14523975	C₂×C₄	1	2	yes	—
76	Dic₆₇⋊C₈	—	2	2	no	—
77	Dic₆₇⋊C₈	—	2	2	no	—
78	Dic₆₇⋊C₈	—	2	1	no	—
79	Dic₆₇⋊C₈	—	2	1	no	—
80	#14523968	C₂×C₄	1	2	yes	—
81	#14523968	C₂×C₄	1	2	yes	—
82	#14523968	C₂×C₄	1	2	yes	—
83	#14523965	C₂×C₄	1	2	yes	—
84	(C₂×D₆₇)⋊C₈	—	2	1	no	—
85	(C₂×D₆₇)⋊C₈	—	2	1	no	—
86	#14523954	—	2	1	no	—
87	#14523953	—	2	2	no	—
88	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
89	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
90	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
91	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
93	C₈×Dic₆₇	—	2	2	no	—
94	C₈×Dic₆₇	—	2	1	no	—
95	C₈×Dic₆₇	—	2	1	no	—
96	C₂×M₁₆×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	1	no	—
99	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
102	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
104	C₂²⋊C₈×C₆₇	—	2	1	no	—
105	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
106	C₄×C₈×C₆₇	—	2	1	no	—
107	C₄×C₈×C₆₇	C₂³	1	2	yes	—
108	#11761628	—	2	2	no	—
109	#11761628	—	2	2	no	—
110	C₂×Dic₁₃₄	—	2	1	no	—
111	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
112	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
113	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₂²⋊D₆₇	—	2	1	no	—
115	C₂×C₄×D₆₇	—	2	1	no	—
116	C₂×C₄×D₆₇	—	2	1	no	—
117	C₂×C₄×D₆₇	—	2	1	no	—
118	Dic₂₆₈	—	2	4	no	—
119	#11761617	—	2	4	no	—
120	#11761612	—	2	4	no	—
121	C₂×C₆₇⋊C₈	—	2	2	no	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	—	4	1	no	—
124	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
128	C₈×D₆₇	—	4	2	no	—
129	C₈×D₆₇	—	4	2	no	—
130	C₄⋊Dic₆₇	—	2	1	no	—
131	C₄⋊Dic₆₇	—	2	1	no	—
132	C₄⋊Dic₆₇	—	2	1	no	—
133	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
134	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
136	Dic₆₇⋊C₄	—	2	1	no	—
137	Dic₆₇⋊C₄	—	2	2	no	—
138	Dic₆₇⋊C₄	—	2	1	no	—
139	Dic₆₇⋊C₄	—	2	2	no	—
140	C₂²⋊Dic₆₇	—	2	2	no	—
141	C₂²⋊Dic₆₇	—	2	1	no	—
142	(C₂×D₆₇)⋊C₄	—	2	1	no	—
143	(C₂×D₆₇)⋊C₄	—	2	1	no	—
144	C₄×Dic₆₇	—	2	1	no	—
145	C₄×Dic₆₇	—	2	1	no	—
146	C₄×Dic₆₇	—	2	1	no	—
147	C₄×Dic₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
151	M₁₆×C₆₇	—	4	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
157	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₆₇	#41	1	1	yes	characteristic
160	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
161	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
162	C₂²×D₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂²⋊D₆₇	—	2	4	no	—
175	C₂×Dic₆₇	—	2	1	no	—
176	C₂×Dic₆₇	—	2	1	no	—
177	C₂×Dic₆₇	—	2	1	no	—
178	C₂×Dic₆₇	—	2	2	no	—
179	C₂×Dic₆₇	—	2	2	no	—
180	D₂₆₈	—	2	1	no	—
181	D₂₆₈	#131	1	2	yes	—
182	C₄×D₆₇	—	2	4	no	—
183	C₄×D₆₇	—	4	1	no	—
184	C₄×D₆₇	—	4	1	no	—
185	C₄×D₆₇	—	4	2	no	—
186	Dic₁₃₄	—	2	1	no	—
187	Dic₁₃₄	—	2	2	no	—
188	Dic₁₃₄	—	2	2	no	—
189	Dic₁₃₄	—	4	2	no	—
190	Dic₁₃₄	#131	1	2	yes	—
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	4	1	no	—
194	C₈×C₆₇	—	4	2	no	—
195	C₈×C₆₇	—	4	1	no	—
196	C₆₇⋊C₈	—	2	4	no	—
197	C₆₇⋊C₈	—	2	1	no	—
198	C₆₇⋊C₈	—	2	1	no	—
199	C₆₇⋊C₈	—	2	1	no	—
200	C₆₇⋊C₈	#137	1	2	yes	—
201	C₂²×C₆₇	—	2	1	no	—
202	C₂²×C₆₇	—	2	1	no	—
203	C₂²×C₆₇	—	2	1	no	—
204	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂×D₆₇	—	2	2	no	—
206	C₂×D₆₇	—	4	1	no	—
207	C₂×D₆₇	—	4	1	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	2	1	no	—
211	C₄×C₆₇	#443	1	2	yes	—
212	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
213	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
214	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
215	C₄×C₆₇	#575	1	1	yes	characteristic
216	Dic₆₇	—	2	2	no	—
217	Dic₆₇	—	4	2	no	—
218	Dic₆₇	—	4	1	no	—
219	Dic₆₇	—	4	2	no	—
220	Dic₆₇	—	4	1	no	—
221	#17913	—	67	1	no	maximal, Sylow
222	C₂×C₆₇	—	4	1	no	—
223	C₂×C₆₇	#2929	1	1	yes	characteristic
224	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
225	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
226	D₆₇	—	4	2	no	—
227	#3107	—	67	1	no	—
228	C₈×#41	—	67	1	no	—
229	#2960	—	67	1	no	—
230	#1587	—	67	2	no	—
231	#1585	—	67	2	no	—
232	C₈×Q₁₆	—	67	2	no	—
233	C₈×SD₁₆	—	67	2	no	—
234	#1574	D₆₇	1	1	yes	characteristic, complemented
235	#1489	—	67	1	no	—
236	#1485	—	67	1	no	—
237	C₈×M₁₆	—	67	1	no	—
238	C₆₇	#17913	1	1	yes	characteristic, complemented, Sylow
239	#533	—	67	1	no	—
240	C₄×#41	—	67	1	no	—
241	#485	—	67	2	no	—
242	#475	—	67	1	no	—
243	#474	—	67	1	no	—
244	#462	—	67	1	no	—
245	#461	—	67	1	no	—
246	#452	—	67	2	no	—
247	#446	—	67	2	no	—
248	#446	—	134	1	no	—
249	C₈×Q₈	—	67	2	no	—
250	C₈×Q₈	—	67	1	no	—
251	C₈×Q₈	—	67	1	no	—
252	C₈×Q₈	—	134	1	no	—
253	C₄×Q₁₆	—	67	2	no	—
254	C₄×SD₁₆	—	67	2	no	—
255	C₈×D₄	—	67	1	no	—
256	C₈×D₄	—	67	1	no	—
257	C₈×D₄	—	134	1	no	—
258	C₈×D₄	—	134	1	no	—
259	#431	—	134	1	no	—
260	#431	C₂×D₆₇	1	1	yes	characteristic
261	#431	C₂×D₆₇	1	1	yes	characteristic
262	#424	—	134	1	no	—
263	#423	C₂×D₆₇	1	1	yes	characteristic
264	#405	—	134	1	no	—
265	C₄×M₁₆	—	67	1	no	—
266	C₂×C₄×C₈	—	67	1	no	—
267	C₂×C₄×C₈	—	134	1	no	—
268	#335	C₂×D₆₇	1	2	yes	—
269	#334	C₂×D₆₇	1	2	yes	—
270	Q₈⋊C₈	—	67	2	no	—
271	Q₈⋊C₈	—	67	2	no	—
272	Q₈⋊C₈	—	67	2	no	—
273	D₄⋊C₈	—	67	2	no	—
274	#322	—	67	2	no	—
275	C₈²	—	67	2	no	—
276	C₂×#41	—	67	1	no	—
277	C₂×Q₁₆	—	67	2	no	—
278	C₂×SD₁₆	—	67	2	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	134	1	no	—
281	C₂²×C₈	—	67	1	no	—
282	C₂²×C₈	—	67	1	no	—
283	C₂²×C₈	—	134	1	no	—
284	C₂²×C₈	—	134	1	no	—
285	C₄⋊Q₈	—	67	1	no	—
286	C₄⋊Q₈	—	67	1	no	—
287	#125	—	134	1	no	—
288	C₂²⋊Q₈	—	67	2	no	—
289	C₂²⋊Q₈	—	134	1	no	—
290	C₄×Q₈	—	67	1	no	—
291	C₄×Q₈	—	67	1	no	—
292	C₄×Q₈	—	67	2	no	—
293	C₄×Q₈	—	134	1	no	—
294	C₄×D₄	—	67	1	no	—
295	C₄×D₄	—	67	1	no	—
296	C₄×D₄	—	134	1	no	—
297	C₄×D₄	—	134	1	no	—
298	#117	—	67	1	no	—
299	#117	—	134	1	no	—
300	#117	—	134	1	no	—
301	C₂×C₄²	—	134	1	no	—
302	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
303	#107	—	67	2	no	—
304	#106	—	67	2	no	—
305	C₄⋊C₈	—	2	1	no	—
306	C₄⋊C₈	—	134	2	no	—
307	C₄⋊C₈	—	134	2	no	—
308	C₄⋊C₈	—	134	1	no	—
309	C₄⋊C₈	—	134	1	no	—
310	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
311	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
312	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
313	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
314	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
315	#103	—	67	2	no	—
316	#103	—	67	2	no	—
317	#103	—	67	2	no	—
318	#102	—	67	2	no	—
319	C₂²⋊C₈	—	2	1	no	—
320	C₂²⋊C₈	—	134	1	no	—
321	C₂²⋊C₈	—	134	1	no	—
322	C₂²⋊C₈	C₄×D₆₇	1	2	yes	—
323	#97	—	134	2	no	—
324	#97	—	134	1	no	—
325	C₄×C₈	—	2	1	no	—
326	C₄×C₈	—	67	1	no	—
327	C₄×C₈	—	67	1	no	—
328	C₄×C₈	—	67	1	no	—
329	C₄×C₈	—	67	1	no	—
330	C₄×C₈	—	67	2	no	—
331	C₄×C₈	—	134	1	no	—
332	C₄×C₈	—	134	1	no	—
333	C₄×C₈	—	134	2	no	—
334	C₄×C₈	C₂²×D₆₇	1	2	yes	—
335	#41	—	134	2	no	—
336	#41	—	134	2	no	—
337	C₂×Q₈	—	67	1	no	—
338	C₂×Q₈	—	67	2	no	—
339	C₂×Q₈	—	134	1	no	—
340	C₂×D₄	—	67	1	no	—
341	C₂×D₄	—	134	1	no	—
342	C₂²×C₄	—	134	1	no	—
343	C₂²×C₄	—	134	1	no	—
344	C₂²×C₄	—	134	1	no	—
345	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
346	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
347	C₂²×C₄	#11761629	1	1	yes	characteristic
348	Q₁₆	—	134	4	no	—
349	SD₁₆	—	134	4	no	—
350	M₁₆	—	4	2	no	—
351	M₁₆	—	134	4	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	268	2	no	—
356	C₂×C₈	—	268	2	no	—
357	C₂×C₈	—	268	1	no	—
358	C₂×C₈	—	67	1	no	—
359	C₂×C₈	—	67	1	no	—
360	C₂×C₈	—	67	1	no	—
361	C₂×C₈	—	67	1	no	—
362	C₂×C₈	—	134	2	no	—
363	C₂×C₈	—	134	2	no	—
364	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
365	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
366	C₄⋊C₄	—	67	2	no	—
367	C₄⋊C₄	—	67	1	no	—
368	C₄⋊C₄	—	67	1	no	—
369	C₄⋊C₄	—	134	1	no	—
370	C₄⋊C₄	—	134	1	no	—
371	C₄⋊C₄	—	134	1	no	—
372	C₄⋊C₄	—	134	2	no	—
373	C₄⋊C₄	—	134	1	no	—
374	C₄⋊C₄	—	134	1	no	—
375	C₄⋊C₄	—	134	2	no	—
376	C₂²⋊C₄	—	134	1	no	—
377	C₂²⋊C₄	—	134	2	no	—
378	C₂²⋊C₄	—	134	1	no	—
379	C₂²⋊C₄	—	134	1	no	—
380	C₄²	—	134	1	no	—
381	C₄²	—	134	1	no	—
382	C₄²	—	134	1	no	—
383	C₄²	—	134	2	no	—
384	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
385	C₄²	D₄×D₆₇	1	1	yes	characteristic
386	C₄²	#11761629	1	1	yes	characteristic
387	C₄²	C₂³×D₆₇	1	1	yes	characteristic
388	C₂³	—	134	1	no	—
389	C₂³	#14524005	1	1	yes	characteristic
390	Q₈	—	268	2	no	—
391	Q₈	—	67	2	no	—
392	Q₈	—	134	2	no	—
393	Q₈	—	134	2	no	—
394	Q₈	—	134	1	no	—
395	D₄	—	67	2	no	—
396	D₄	—	134	4	no	—
397	D₄	—	134	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	268	2	no	—
404	C₂×C₄	—	268	1	no	—
405	C₂×C₄	—	268	1	no	—
406	C₂×C₄	—	134	4	no	—
407	C₂×C₄	—	134	1	no	—
408	C₂×C₄	—	134	1	no	—
409	C₂×C₄	—	134	2	no	—
410	C₂×C₄	—	134	2	no	—
411	C₂×C₄	—	134	1	no	—
412	C₂×C₄	#14524005	1	1	yes	characteristic
413	C₂×C₄	#14524005	1	1	yes	characteristic
414	C₂×C₄	#14524005	1	1	yes	characteristic
415	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
417	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	1	no	—
421	C₈	—	4	1	no	—
422	C₈	—	4	2	no	—
423	C₈	—	67	2	no	—
424	C₈	—	134	4	no	—
425	C₈	—	134	1	no	—
426	C₈	—	134	1	no	—
427	C₈	—	134	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	268	1	no	—
432	C₂²	—	268	1	no	—
433	C₂²	—	134	2	no	—
434	C₂²	C₂×#14524005	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	268	1	no	—
439	C₄	—	268	1	no	—
440	C₄	—	268	2	no	—
441	C₄	—	268	2	no	—
442	C₄	—	134	2	no	—
443	C₄	#16863512	1	2	yes	—
444	C₄	#131×D₆₇	1	1	yes	characteristic
445	C₄	#131×D₆₇	1	1	yes	characteristic
446	C₄	#137×D₆₇	1	1	yes	characteristic
447	C₄	#16864192	1	1	yes	characteristic
448	C₂	—	4	1	no	—
449	C₂	—	268	2	no	—
450	C₂	#22568902	1	1	yes	characteristic
451	C₂	#22569027	1	1	yes	characteristic
452	C₂	C₂×#16863512	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×#1574	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1574	D₆₇
hypocenter	C₆₇	#17913
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1574

Complements

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

N	H ≅ G/N	type
3. #22569139	C₂	semidirect
4. #22559079	C₂	semidirect
5. #22559065	C₂	semidirect
6. #22559015	C₂	semidirect
7. #22559008	C₂	semidirect
9. #22558629	C₂	semidirect
10. #22558490	C₂	semidirect
11. C₆₇×#1574	C₂	semidirect
234. #1574	D₆₇	semidirect
238. C₆₇	#17913	semidirect

Maximal subgroups

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

subgroup	index
#22570281	2
#22569165	2
#22569139	2
#22559079	2
#22559065	2
#22559015	2
#22559008	2
#22558646	2
#22558629	2
#22558490	2
C₆₇×#1574	2
#17913	67