Subgroups — #30874427

Order 17152.

916 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863474	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560394	C₂	1	2	yes	maximal, complemented
6	#22560389	C₂	1	2	yes	maximal, complemented
7	#22560326	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560193	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559906	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2293	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11761612	C₂²	1	2	yes	—
13	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
14	#16863474	—	2	4	no	—
15	D₄×C₆₇⋊C₈	—	2	4	no	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
22	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
23	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
24	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	#16863151	C₂²	1	1	yes	characteristic
27	#16863151	C₂²	1	1	yes	characteristic
28	#16863151	C₂²	1	2	yes	—
29	#16863151	C₂²	1	2	yes	—
30	#16863151	C₂²	1	2	yes	—
31	#16863151	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#117×C₆₇	C₂²	1	2	yes	—
34	#390×C₆₇	C₄	1	1	yes	characteristic
35	#390×C₆₇	C₄	1	1	yes	characteristic
36	#389×C₆₇	C₄	1	1	yes	characteristic
37	#389×C₆₇	C₄	1	1	yes	characteristic
38	#388×C₆₇	C₄	1	2	yes	—
39	#388×C₆₇	C₄	1	2	yes	—
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
42	C₄³×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#11761612	—	2	4	no	—
44	C₂×#11761612	—	2	4	no	—
45	C₂×#11761612	—	4	2	no	—
46	C₂²×C₆₇⋊C₈	—	2	4	no	—
47	C₂²×C₆₇⋊C₈	—	2	4	no	—
48	C₂²×C₆₇⋊C₈	—	4	2	no	—
49	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
57	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
59	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
60	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
61	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
62	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
63	C₄×C₆₇⋊C₈	—	2	2	no	—
64	C₄×C₆₇⋊C₈	—	2	2	no	—
65	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₄×D₄×C₆₇	D₄	1	4	yes	—
68	C₄×D₄×C₆₇	Q₈	1	4	yes	—
69	#117×C₆₇	—	4	2	no	—
70	#117×C₆₇	—	4	2	no	—
71	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	#95×C₆₇	C₂×C₄	1	2	yes	—
90	#95×C₆₇	C₂×C₄	1	2	yes	—
91	#95×C₆₇	C₂×C₄	1	2	yes	—
92	#11761612	—	4	8	no	—
93	C₂×C₆₇⋊C₈	—	2	1	no	—
94	C₂×C₆₇⋊C₈	—	2	1	no	—
95	C₂×C₆₇⋊C₈	—	2	1	no	—
96	C₂×C₆₇⋊C₈	—	2	1	no	—
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	—	2	1	no	—
99	C₂×C₆₇⋊C₈	—	2	2	no	—
100	C₂×C₆₇⋊C₈	—	2	2	no	—
101	C₂×C₆₇⋊C₈	—	2	1	no	—
102	C₂×C₆₇⋊C₈	—	2	1	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	1	no	—
106	C₂×C₆₇⋊C₈	—	2	1	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	—	2	2	no	—
109	C₂×C₆₇⋊C₈	—	4	8	no	—
110	C₂×C₆₇⋊C₈	#41	1	2	yes	—
111	C₂×C₆₇⋊C₈	#41	1	2	yes	—
112	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
131	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	2	no	—
136	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	4	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	4	2	no	—
141	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	4	1	no	—
152	C₄²×C₆₇	C₂×D₄	1	2	yes	—
153	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
161	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
162	D₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
190	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₄×C₆₇	#131	1	2	yes	—
192	C₂×C₄×C₆₇	#131	1	2	yes	—
193	C₂×C₄×C₆₇	#131	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
199	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
200	C₆₇⋊C₈	—	2	2	no	—
201	C₆₇⋊C₈	—	2	2	no	—
202	C₆₇⋊C₈	—	4	1	no	—
203	C₆₇⋊C₈	—	4	1	no	—
204	C₆₇⋊C₈	—	4	2	no	—
205	C₆₇⋊C₈	—	4	2	no	—
206	C₂²×C₆₇	—	2	4	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	2	4	no	—
209	C₂²×C₆₇	—	4	2	no	—
210	C₂²×C₆₇	—	4	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
215	C₂²×C₆₇	#518	1	1	yes	characteristic
216	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
218	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
219	C₂²×C₆₇	#568	1	1	yes	characteristic
220	C₄×C₆₇	—	2	4	no	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	4	1	no	—
224	C₄×C₆₇	—	4	1	no	—
225	C₄×C₆₇	—	4	2	no	—
226	C₄×C₆₇	—	4	1	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	#424	1	4	yes	—
229	C₄×C₆₇	#546	1	1	yes	characteristic
230	C₄×C₆₇	#546	1	1	yes	characteristic
231	C₄×C₆₇	#554	1	1	yes	characteristic
232	C₄×C₆₇	#554	1	1	yes	characteristic
233	#22701	—	67	1	no	maximal, Sylow
234	C₂×C₆₇	—	4	4	no	—
235	C₂×C₆₇	#2349	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
237	C₂×C₆₇	#2906	1	1	yes	characteristic
238	C₂×C₆₇	#2966	1	1	yes	characteristic
239	C₂×C₆₇	#2969	1	1	yes	characteristic
240	C₂×C₆₇	#2979	1	1	yes	characteristic
241	C₂×C₆₇	#2981	1	1	yes	characteristic
242	C₂×#436	—	67	1	no	—
243	C₂×C₈×D₄	—	67	1	no	—
244	#2293	D₆₇	1	1	yes	characteristic
245	#1939	—	67	1	no	—
246	#1939	—	67	1	no	—
247	#1851	—	67	2	no	—
248	#1850	—	67	2	no	—
249	#1831	—	67	1	no	—
250	#1798	—	67	2	no	—
251	C₄×C₄⋊C₈	—	67	1	no	—
252	#1729	—	67	2	no	—
253	C₆₇	#22701	1	1	yes	characteristic, complemented, Sylow
254	C₂²×M₁₆	—	67	2	no	—
255	C₂³×C₈	—	67	2	no	—
256	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
257	C₂×#117	C₂×D₆₇	1	2	yes	—
258	#436	—	134	4	no	—
259	C₈×D₄	—	134	4	no	—
260	C₂×C₄⋊C₈	—	67	2	no	—
261	C₂×C₄⋊C₈	—	67	2	no	—
262	C₂×C₄⋊C₈	—	67	1	no	—
263	C₂×C₄⋊C₈	—	67	1	no	—
264	C₂×C₄⋊C₈	—	67	1	no	—
265	C₂×C₄⋊C₈	—	67	1	no	—
266	C₂×C₂²⋊C₈	—	67	2	no	—
267	C₂×C₂²⋊C₈	—	67	2	no	—
268	C₂×C₄×C₈	—	67	1	no	—
269	C₂×C₄×C₈	—	67	1	no	—
270	#390	Dic₆₇	1	1	yes	characteristic
271	#390	Dic₆₇	1	1	yes	characteristic
272	#389	Dic₆₇	1	1	yes	characteristic
273	#389	Dic₆₇	1	1	yes	characteristic
274	#388	Dic₆₇	1	2	yes	—
275	#388	Dic₆₇	1	2	yes	—
276	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
277	C₄×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
278	C₄³	C₂×D₆₇	1	1	yes	characteristic
279	#336	—	67	1	no	—
280	#336	—	67	2	no	—
281	#336	—	67	2	no	—
282	#336	—	67	2	no	—
283	#336	—	67	2	no	—
284	#336	—	67	1	no	—
285	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
286	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
287	C₂×M₁₆	—	268	2	no	—
288	C₂×M₁₆	—	134	4	no	—
289	C₂×M₁₆	—	134	4	no	—
290	C₂²×C₈	—	268	2	no	—
291	C₂²×C₈	—	67	1	no	—
292	C₂²×C₈	—	67	1	no	—
293	C₂²×C₈	—	67	2	no	—
294	C₂²×C₈	—	67	2	no	—
295	C₂²×C₈	—	67	1	no	—
296	C₂²×C₈	—	67	1	no	—
297	C₂²×C₈	—	134	4	no	—
298	C₂²×C₈	—	134	4	no	—
299	C₄×D₄	Dic₁₃₄	1	4	yes	—
300	C₄×D₄	D₂₆₈	1	4	yes	—
301	#117	—	4	2	no	—
302	#117	—	4	2	no	—
303	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
306	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
307	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
309	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
310	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
311	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
312	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
313	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
314	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
315	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
316	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
317	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
318	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
319	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
321	C₄⋊C₈	—	268	2	no	—
322	C₄⋊C₈	—	268	2	no	—
323	C₄⋊C₈	—	134	2	no	—
324	C₄⋊C₈	—	134	2	no	—
325	C₄⋊C₈	—	134	2	no	—
326	C₄⋊C₈	—	134	2	no	—
327	C₂²⋊C₈	—	134	4	no	—
328	C₂²⋊C₈	—	134	4	no	—
329	C₄×C₈	—	134	2	no	—
330	C₄×C₈	—	134	2	no	—
331	#95	C₂×Dic₆₇	1	2	yes	—
332	#95	C₂×Dic₆₇	1	2	yes	—
333	#95	C₂×Dic₆₇	1	2	yes	—
334	C₂⁴	C₂²×Dic₆₇	1	2	yes	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	C₄⋊Dic₆₇	1	4	yes	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
343	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
344	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
345	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
349	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
352	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
353	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
354	M₁₆	—	268	8	no	—
355	C₂×C₈	—	268	8	no	—
356	C₂×C₈	—	67	2	no	—
357	C₂×C₈	—	67	2	no	—
358	C₂×C₈	—	134	1	no	—
359	C₂×C₈	—	134	1	no	—
360	C₂×C₈	—	134	2	no	—
361	C₂×C₈	—	134	2	no	—
362	C₂×C₈	—	134	1	no	—
363	C₂×C₈	—	134	1	no	—
364	C₂×C₈	—	134	2	no	—
365	C₂×C₈	—	134	2	no	—
366	C₂×C₈	—	134	1	no	—
367	C₂×C₈	—	134	1	no	—
368	C₂×C₈	—	134	2	no	—
369	C₂×C₈	—	134	2	no	—
370	C₂×C₈	—	134	1	no	—
371	C₂×C₈	—	134	1	no	—
372	C₂×C₈	—	134	1	no	—
373	C₂×C₈	—	134	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	1	no	—
392	C₄²	—	2	1	no	—
393	C₄²	—	4	1	no	—
394	C₄²	C₂×D₂₆₈	1	2	yes	—
395	C₄²	C₂×Dic₁₃₄	1	2	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
403	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
404	D₄	—	2	8	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
431	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
432	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
433	C₂×C₄	#14524050	1	2	yes	—
434	C₂×C₄	#14524050	1	2	yes	—
435	C₂×C₄	#14524050	1	2	yes	—
436	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
437	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
438	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
439	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
440	C₂×C₄	#14524088	1	1	yes	characteristic
441	C₂×C₄	#14524088	1	1	yes	characteristic
442	C₈	—	268	2	no	—
443	C₈	—	268	2	no	—
444	C₈	—	268	1	no	—
445	C₈	—	268	1	no	—
446	C₈	—	134	2	no	—
447	C₈	—	134	2	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
456	C₂²	#41×Dic₆₇	1	1	yes	characteristic
457	C₂²	#16863871	1	1	yes	characteristic
458	C₂²	C₂×#14524050	1	1	yes	characteristic
459	C₂²	C₂×#14524050	1	1	yes	characteristic
460	C₂²	C₂×#14524050	1	1	yes	characteristic
461	C₂²	#16864163	1	1	yes	characteristic
462	C₄	—	2	4	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	1	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	1	no	—
470	C₄	#16863423	1	4	yes	—
471	C₄	#16864041	1	1	yes	characteristic
472	C₄	#16864041	1	1	yes	characteristic
473	C₄	#16864105	1	1	yes	characteristic
474	C₄	#16864105	1	1	yes	characteristic
475	C₂	—	4	4	no	—
476	C₂	#22563119	1	1	yes	characteristic
477	C₂	C₂×#16863423	1	1	yes	characteristic
478	C₂	#22568777	1	1	yes	characteristic
479	C₂	#22569185	1	1	yes	characteristic
480	C₂	#22569214	1	1	yes	characteristic
481	C₂	#22569276	1	1	yes	characteristic
482	C₂	#22569285	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2293	C₂
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2293	D₆₇
hypocenter	C₆₇	#22701
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2293

Complements

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

N	H ≅ G/N	type
3. #22560928	C₂	semidirect
4. #22560928	C₂	semidirect
5. #22560394	C₂	semidirect
6. #22560389	C₂	semidirect
7. #22560326	C₂	semidirect
8. #22560193	C₂	semidirect
9. C₄×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
10. #22559906	C₂	semidirect
253. C₆₇	#22701	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863474	2
C₂×D₄×C₆₇⋊C₈	2
#22560928	2
#22560394	2
#22560389	2
#22560326	2
#22560193	2
C₄×C₄⋊(C₆₇⋊C₈)	2
#22559906	2
C₆₇×#2293	2
#22701	67