Subgroups — #30748520

Order 17152.

588 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22560361	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863128	C₂	1	2	yes	maximal
4	#22558049	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₄×Dic₆₇⋊C₄	—	2	1	no	—
13	C₄×Dic₆₇⋊C₄	—	2	1	no	—
14	C₄×Dic₆₇⋊C₄	—	2	1	no	—
15	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
16	C₄×C₄⋊Dic₆₇	C₄	1	1	yes	characteristic
17	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
18	C₄²×Dic₆₇	—	2	1	no	—
19	#16863128	—	2	4	no	—
20	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	#336×C₆₇	—	2	1	no	—
24	C₂²×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₄×Dic₁₃₄	—	2	2	no	—
27	C₄×Dic₁₃₄	—	2	2	no	—
28	C₄×Dic₁₃₄	—	2	2	no	—
29	C₄×Dic₁₃₄	—	2	1	no	—
30	C₄×Dic₁₃₄	—	2	1	no	—
31	C₄×Dic₁₃₄	C₈	1	4	yes	complemented
32	C₄×Dic₁₃₄	D₄	1	2	yes	—
33	C₄×Dic₁₃₄	Q₈	1	2	yes	—
34	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₆₇⋊C₄	—	2	1	no	—
38	C₂×Dic₆₇⋊C₄	—	2	1	no	—
39	C₂×Dic₆₇⋊C₄	—	2	1	no	—
40	C₂×C₄×Dic₆₇	—	2	1	no	—
41	C₂×C₄×Dic₆₇	—	2	1	no	—
42	C₂×C₄×Dic₆₇	—	2	1	no	—
43	C₂×C₄×Dic₆₇	—	2	1	no	—
44	C₂×C₄×Dic₆₇	—	2	1	no	—
45	C₂×C₄×Dic₆₇	—	2	1	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
48	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
49	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
50	C₆₇⋊#95	—	2	1	no	—
51	C₆₇⋊#95	—	2	1	no	—
52	C₆₇⋊#95	—	2	1	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₆₇	—	2	4	no	—
62	C₄×C₈×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	—	2	2	no	—
64	C₂×Dic₁₃₄	—	2	1	no	—
65	C₂×Dic₁₃₄	—	2	1	no	—
66	C₂×Dic₁₃₄	—	2	1	no	—
67	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
68	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
69	C₂×Dic₁₃₄	M₁₆	1	2	yes	—
70	C₂²×Dic₆₇	—	2	1	no	—
71	C₂²×Dic₆₇	—	2	1	no	—
72	C₂²×Dic₆₇	—	2	1	no	—
73	C₂²×Dic₆₇	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	4	2	no	—
75	C₂×C₆₇⋊C₈	—	4	2	no	—
76	C₂×C₆₇⋊C₈	—	4	2	no	—
77	C₄⋊Dic₆₇	—	2	1	no	—
78	C₄⋊Dic₆₇	—	2	1	no	—
79	C₄⋊Dic₆₇	—	2	1	no	—
80	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
81	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
82	C₄⋊Dic₆₇	M₁₆	1	2	yes	—
83	Dic₆₇⋊C₄	—	2	4	no	—
84	Dic₆₇⋊C₄	—	2	4	no	—
85	Dic₆₇⋊C₄	—	2	2	no	—
86	Dic₆₇⋊C₄	—	2	2	no	—
87	C₄×Dic₆₇	—	2	4	no	—
88	C₄×Dic₆₇	—	2	4	no	—
89	C₄×Dic₆₇	—	2	2	no	—
90	C₄×Dic₆₇	—	2	2	no	—
91	C₄×Dic₆₇	—	4	2	no	—
92	C₄×Dic₆₇	—	4	2	no	—
93	C₄×Dic₆₇	—	4	1	no	—
94	C₄×Dic₆₇	—	4	1	no	—
95	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	4	2	no	—
104	C₂×C₈×C₆₇	—	4	2	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	—	2	1	no	—
109	C₄²×C₆₇	—	2	1	no	—
110	C₄²×C₆₇	—	4	1	no	—
111	C₄²×C₆₇	C₂×C₈	1	2	yes	—
112	C₄²×C₆₇	M₁₆	1	2	yes	—
113	C₄²×C₆₇	SD₁₆	1	2	yes	—
114	C₄²×C₆₇	Q₁₆	1	2	yes	—
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₆₇	#41	1	1	yes	characteristic
118	C₄²×C₆₇	#41	1	1	yes	characteristic
119	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
130	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
131	C₂×C₄×C₆₇	#103	1	2	yes	—
132	C₂×C₄×C₆₇	#103	1	2	yes	—
133	C₂×C₄×C₆₇	#104	1	2	yes	—
134	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
146	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	2	2	no	—
151	C₂×Dic₆₇	—	2	2	no	—
152	C₂×Dic₆₇	—	2	2	no	—
153	C₂×Dic₆₇	—	2	2	no	—
154	C₂×Dic₆₇	—	4	1	no	—
155	C₂×Dic₆₇	—	4	1	no	—
156	C₂×Dic₆₇	—	4	1	no	—
157	C₂×Dic₆₇	—	4	1	no	—
158	C₂×Dic₆₇	—	4	1	no	—
159	C₂×Dic₆₇	—	4	1	no	—
160	Dic₁₃₄	—	2	2	no	—
161	Dic₁₃₄	—	2	2	no	—
162	Dic₁₃₄	—	2	2	no	—
163	Dic₁₃₄	C₄⋊C₈	1	4	yes	complemented
164	C₈×C₆₇	—	4	4	no	—
165	C₆₇⋊C₈	—	8	2	no	—
166	C₂²×C₆₇	#380	1	1	yes	characteristic
167	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂²×C₆₇	#407	1	1	yes	characteristic
169	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
170	C₂²×C₆₇	#417	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
172	C₂²×C₆₇	#421	1	1	yes	characteristic
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	4	1	no	—
178	C₄×C₆₇	Q₈⋊C₈	1	4	yes	complemented
179	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
180	C₄×C₆₇	#423	1	1	yes	characteristic
181	C₄×C₆₇	#431	1	1	yes	characteristic
182	C₄×C₆₇	#432	1	1	yes	characteristic
183	C₄×C₆₇	#460	1	1	yes	characteristic
184	C₄×C₆₇	#462	1	1	yes	characteristic
185	C₄×C₆₇	#475	1	1	yes	characteristic
186	C₄×C₆₇	#477	1	1	yes	characteristic
187	Dic₆₇	—	2	4	no	—
188	Dic₆₇	—	4	2	no	—
189	Dic₆₇	—	4	2	no	—
190	Dic₆₇	—	4	2	no	—
191	#8564	—	67	1	no	maximal, Sylow
192	C₂×C₆₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
193	C₂×C₆₇	#1477	1	1	yes	characteristic
194	C₂×C₆₇	#1487	1	1	yes	characteristic
195	C₂×C₆₇	#1489	1	1	yes	characteristic
196	C₂×C₆₇	#1797	1	1	yes	characteristic
197	C₂×C₆₇	#1803	1	1	yes	characteristic
198	C₂×C₆₇	#1806	1	1	yes	characteristic
199	C₄²×Q₈	—	67	1	no	—
200	#1842	—	67	1	no	—
201	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
202	C₂×Q₈⋊C₈	—	67	2	no	—
203	#1279	—	67	2	no	—
204	C₆₇	#8564	1	1	yes	characteristic, complemented, Sylow
205	C₂×C₄×Q₈	—	67	1	no	—
206	C₂×C₄×Q₈	—	67	1	no	—
207	C₂×C₄×Q₈	—	67	1	no	—
208	C₂×C₄⋊C₈	—	2	1	no	—
209	C₂×C₄⋊C₈	—	67	2	no	—
210	C₂×C₄⋊C₈	—	134	1	no	—
211	C₂×C₄⋊C₈	—	134	1	no	—
212	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
213	C₄×C₄⋊C₄	—	67	1	no	—
214	C₄×C₄⋊C₄	—	67	1	no	—
215	C₄×C₄⋊C₄	—	67	1	no	—
216	C₄×C₄⋊C₄	—	134	1	no	—
217	C₄×C₄⋊C₄	—	134	1	no	—
218	C₄×C₄⋊C₄	—	134	1	no	—
219	C₄³	—	134	1	no	—
220	C₄³	C₂×D₆₇	1	1	yes	characteristic
221	#336	—	2	1	no	—
222	Q₈⋊C₈	—	134	4	no	—
223	C₂²×Q₈	—	67	1	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	134	2	no	—
226	C₄×Q₈	—	67	4	no	—
227	C₄×Q₈	—	67	2	no	—
228	C₄×Q₈	—	67	2	no	—
229	C₄×Q₈	—	134	2	no	—
230	C₄×Q₈	—	134	2	no	—
231	C₄×Q₈	—	134	2	no	—
232	C₄×Q₈	—	134	1	no	—
233	C₄×Q₈	—	134	1	no	—
234	C₂×C₄⋊C₄	—	67	1	no	—
235	C₂×C₄⋊C₄	—	67	1	no	—
236	C₂×C₄⋊C₄	—	67	1	no	—
237	C₂×C₄⋊C₄	—	134	1	no	—
238	C₂×C₄⋊C₄	—	134	1	no	—
239	C₂×C₄⋊C₄	—	134	1	no	—
240	C₂×C₄²	—	134	1	no	—
241	C₂×C₄²	—	134	1	no	—
242	C₂×C₄²	—	134	1	no	—
243	C₂×C₄²	—	134	1	no	—
244	C₂×C₄²	—	134	1	no	—
245	C₂×C₄²	—	134	1	no	—
246	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
249	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
250	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	268	2	no	—
255	C₄⋊C₈	—	268	2	no	—
256	C₄⋊C₈	—	134	2	no	—
257	C₄⋊C₈	—	134	2	no	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	2	no	—
260	#95	—	134	1	no	—
261	#95	—	134	1	no	—
262	#95	—	134	1	no	—
263	C₂×Q₈	—	67	2	no	—
264	C₂×Q₈	—	67	2	no	—
265	C₂×Q₈	—	67	2	no	—
266	C₂×Q₈	—	134	1	no	—
267	C₂×Q₈	—	134	1	no	—
268	C₂×Q₈	—	134	1	no	—
269	C₂²×C₄	—	134	1	no	—
270	C₂²×C₄	—	134	1	no	—
271	C₂²×C₄	—	134	1	no	—
272	C₂²×C₄	—	134	1	no	—
273	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
274	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
275	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
276	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	4	2	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	268	2	no	—
284	C₂×C₈	—	268	2	no	—
285	C₂×C₈	—	268	2	no	—
286	C₄⋊C₄	—	67	2	no	—
287	C₄⋊C₄	—	67	2	no	—
288	C₄⋊C₄	—	67	2	no	—
289	C₄⋊C₄	—	134	4	no	—
290	C₄⋊C₄	—	134	4	no	—
291	C₄⋊C₄	—	134	1	no	—
292	C₄⋊C₄	—	134	2	no	—
293	C₄⋊C₄	—	134	2	no	—
294	C₄⋊C₄	—	134	1	no	—
295	C₄⋊C₄	—	134	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	268	2	no	—
303	C₄²	—	268	2	no	—
304	C₄²	—	268	1	no	—
305	C₄²	—	268	1	no	—
306	C₄²	—	134	4	no	—
307	C₄²	—	134	4	no	—
308	C₄²	—	134	2	no	—
309	C₄²	—	134	2	no	—
310	C₄²	C₈×D₆₇	1	2	yes	—
311	C₄²	#11761611	1	2	yes	—
312	C₄²	#11761617	1	2	yes	—
313	C₄²	Dic₂₆₈	1	2	yes	—
314	C₄²	D₄×D₆₇	1	1	yes	characteristic
315	C₄²	Q₈×D₆₇	1	1	yes	characteristic
316	C₄²	#11761627	1	1	yes	characteristic
317	C₄²	#11761629	1	1	yes	characteristic
318	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
319	Q₈	—	67	4	no	—
320	Q₈	—	134	2	no	—
321	Q₈	—	134	2	no	—
322	Q₈	—	134	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	268	1	no	—
333	C₂×C₄	—	268	1	no	—
334	C₂×C₄	—	268	1	no	—
335	C₂×C₄	—	268	1	no	—
336	C₂×C₄	—	268	1	no	—
337	C₂×C₄	—	268	1	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	—	134	2	no	—
340	C₂×C₄	—	134	2	no	—
341	C₂×C₄	—	134	2	no	—
342	C₂×C₄	—	134	2	no	—
343	C₂×C₄	—	134	2	no	—
344	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
345	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
346	C₂×C₄	#14523968	1	2	yes	—
347	C₂×C₄	#14523968	1	2	yes	—
348	C₂×C₄	#14523971	1	2	yes	—
349	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
350	C₂×C₄	#14524001	1	1	yes	characteristic
351	C₂×C₄	#14524014	1	1	yes	characteristic
352	C₂×C₄	#14524018	1	1	yes	characteristic
353	C₂×C₄	#14524019	1	1	yes	characteristic
354	C₂×C₄	#14524023	1	1	yes	characteristic
355	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
356	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
357	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
359	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
360	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
361	C₂×C₄	#14524071	1	1	yes	characteristic
362	C₂×C₄	#14524076	1	1	yes	characteristic
363	C₈	—	4	4	no	—
364	C₈	—	536	2	no	—
365	C₂²	#16863253	1	1	yes	characteristic
366	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
367	C₂²	#16863359	1	1	yes	characteristic
368	C₂²	C₂×#14523968	1	1	yes	characteristic
369	C₂²	#16863398	1	1	yes	characteristic
370	C₂²	C₂×#14523971	1	1	yes	characteristic
371	C₂²	#16863414	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	268	2	no	—
378	C₄	—	268	2	no	—
379	C₄	—	268	2	no	—
380	C₄	—	134	4	no	—
381	C₄	#16863128	1	4	yes	complemented
382	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
383	C₄	#16863418	1	1	yes	characteristic
384	C₄	#16863448	1	1	yes	characteristic
385	C₄	#16863452	1	1	yes	characteristic
386	C₄	#16863597	1	1	yes	characteristic
387	C₄	#16863607	1	1	yes	characteristic
388	C₄	#16863682	1	1	yes	characteristic
389	C₄	#16863692	1	1	yes	characteristic
390	C₂	C₂×#16863128	1	1	yes	characteristic
391	C₂	#22558597	1	1	yes	characteristic
392	C₂	#22558639	1	1	yes	characteristic
393	C₂	#22558649	1	1	yes	characteristic
394	C₂	#22560186	1	1	yes	characteristic
395	C₂	#22560215	1	1	yes	characteristic
396	C₂	#22560230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#8564
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
31. C₄×Dic₁₃₄	C₈	semidirect
163. Dic₁₃₄	C₄⋊C₈	semidirect
178. C₄×C₆₇	Q₈⋊C₈	semidirect
204. C₆₇	#8564	semidirect
381. C₄	#16863128	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₁₃₄	2
#22560361	2
C₂×#16863128	2
#22558049	2
C₄×C₄⋊C₈×C₆₇	2
#8564	67