Subgroups — C₂×#22558050 #30745356

Order 17152.

974 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558050	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	—	2	1	no	—
6	C₂³×C₆₇⋊C₈	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₆₇⋊C₈	—	2	2	no	—
12	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
13	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	—	2	1	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×#95×C₆₇	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	8	no	—
24	C₂²×C₆₇⋊C₈	—	2	4	no	—
25	C₂²×C₆₇⋊C₈	—	2	4	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	—	4	2	no	—
29	C₂²×C₆₇⋊C₈	—	4	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	complemented
33	C₄×C₆₇⋊C₈	—	2	4	no	—
34	C₄×C₆₇⋊C₈	—	2	2	no	—
35	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
36	C₂³×C₄×C₆₇	—	2	2	no	—
37	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	C₈	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	—	4	2	no	—
47	C₂×C₄²×C₆₇	—	4	2	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#95×C₆₇	—	2	2	no	—
57	#95×C₆₇	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	16	no	—
59	C₂×C₆₇⋊C₈	—	2	8	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	4	8	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	4	4	no	—
70	C₂²×C₄×C₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
81	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
82	C₂²×C₄×C₆₇	D₈	1	2	yes	—
83	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
84	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
85	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₆₇	—	2	8	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	complemented
92	C₄²×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	—	4	2	no	—
94	C₄²×C₆₇	—	4	2	no	—
95	C₄²×C₆₇	C₄²	1	2	yes	—
96	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
99	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
100	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
105	C₂³×C₆₇	#95	1	2	yes	—
106	C₂³×C₆₇	C₂²⋊C₈	1	4	yes	—
107	C₂³×C₆₇	#102	1	2	yes	—
108	C₂³×C₆₇	#103	1	2	yes	—
109	C₂³×C₆₇	#104	1	4	yes	—
110	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	—	2	4	no	—
112	C₂×C₄×C₆₇	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	4	16	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	#95	1	2	yes	—
120	C₂×C₄×C₆₇	#95	1	2	yes	—
121	C₂×C₄×C₆₇	#95	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
123	C₂×C₄×C₆₇	#97	1	4	yes	—
124	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
125	C₂×C₄×C₆₇	#102	1	4	yes	—
126	C₂×C₄×C₆₇	#102	1	2	yes	—
127	C₂×C₄×C₆₇	#103	1	4	yes	—
128	C₂×C₄×C₆₇	#103	1	2	yes	—
129	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
130	C₂×C₄×C₆₇	#106	1	4	yes	—
131	C₂×C₄×C₆₇	#107	1	4	yes	—
132	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
144	C₆₇⋊C₈	—	2	8	no	—
145	C₆₇⋊C₈	—	4	4	no	—
146	C₂²×C₆₇	D₄⋊C₈	1	8	yes	—
147	C₂²×C₆₇	Q₈⋊C₈	1	8	yes	—
148	C₂²×C₆₇	#336	1	4	yes	—
149	C₂²×C₆₇	#339	1	4	yes	—
150	C₂²×C₆₇	#340	1	4	yes	—
151	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
153	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
154	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
155	C₂²×C₆₇	C₂×#104	1	2	yes	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	4	8	no	—
160	C₄×C₆₇	#336	1	4	yes	—
161	C₄×C₆₇	#340	1	4	yes	—
162	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
164	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
165	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
166	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
167	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
168	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
169	C₂×#1279	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	#1279	1	8	yes	complemented
171	C₂×C₆₇	C₂×D₄⋊C₈	1	2	yes	—
172	C₂×C₆₇	C₂×Q₈⋊C₈	1	2	yes	—
173	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
174	C₂×C₆₇	C₂×#339	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
176	C₂²×C₄⋊C₈	—	67	1	no	—
177	C₂²×C₄×C₈	—	67	1	no	—
178	C₂×C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
179	#1279	—	67	4	no	—
180	C₆₇	C₂×#1279	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₈	—	134	1	no	—
182	C₂³×C₈	—	134	1	no	—
183	C₂²×C₄⋊C₄	Dic₆₇	1	2	yes	—
184	C₂²×C₄²	—	2	1	no	—
185	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	67	4	no	—
187	C₂×C₄⋊C₈	—	67	2	no	—
188	C₂×C₄⋊C₈	—	134	2	no	—
189	C₂×C₄⋊C₈	—	134	1	no	—
190	C₂×C₄×C₈	—	67	4	no	—
191	C₂×C₄×C₈	—	67	2	no	—
192	C₂×C₄×C₈	—	134	2	no	—
193	C₂×C₄×C₈	—	134	1	no	—
194	C₄×C₄⋊C₄	—	2	1	no	—
195	C₄×C₄⋊C₄	—	2	1	no	—
196	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
197	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
198	C₂×#95	—	2	1	no	—
199	C₂³×C₄	—	2	2	no	—
200	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
203	C₂²×C₈	—	268	2	no	—
204	C₂²×C₈	—	268	2	no	—
205	C₂²×C₈	—	134	4	no	—
206	C₂²×C₈	—	134	2	no	—
207	C₂²×C₈	—	134	8	no	—
208	C₂²×C₈	—	134	4	no	—
209	C₂²×C₈	—	134	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	C₆₇⋊C₈	1	8	yes	complemented
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	4	2	no	—
217	C₂×C₄²	—	4	2	no	—
218	C₂×C₄²	Dic₁₃₄	1	2	yes	—
219	C₂×C₄²	C₄×D₆₇	1	2	yes	—
220	C₂×C₄²	C₄×D₆₇	1	2	yes	—
221	C₂×C₄²	D₂₆₈	1	2	yes	—
222	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
226	C₄⋊C₈	—	67	4	no	—
227	C₄⋊C₈	—	134	4	no	—
228	C₄⋊C₈	—	134	2	no	—
229	C₄×C₈	—	67	4	no	—
230	C₄×C₈	—	134	4	no	—
231	C₄×C₈	—	134	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
241	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
242	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
243	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
246	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
247	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
248	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
249	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
250	C₂²×C₄	#11761612	1	4	yes	—
251	C₂²×C₄	#11761613	1	2	yes	—
252	C₂²×C₄	#11761615	1	2	yes	—
253	C₂²×C₄	#11761616	1	2	yes	—
254	C₂²×C₄	#11761618	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂×C₈	—	268	8	no	—
259	C₂×C₈	—	268	2	no	—
260	C₂×C₈	—	268	2	no	—
261	C₂×C₈	—	134	4	no	—
262	C₂×C₈	—	134	8	no	—
263	C₂×C₈	—	134	16	no	—
264	C₂×C₈	—	134	4	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	complemented
268	C₄²	—	4	4	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₄×Dic₆₇	1	2	yes	—
272	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
273	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
274	C₄²	Dic₆₇⋊C₄	1	2	yes	—
275	C₄²	Dic₆₇⋊C₄	1	2	yes	—
276	C₄²	C₄⋊Dic₆₇	1	2	yes	—
277	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
278	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
279	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
280	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
281	C₂³	C₆₇⋊#95	1	2	yes	—
282	C₂³	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
283	C₂³	#14523964	1	2	yes	—
284	C₂³	#14523967	1	2	yes	—
285	C₂³	#14523970	1	4	yes	—
286	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	4	16	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	C₆₇⋊#95	1	2	yes	—
296	C₂×C₄	C₆₇⋊#95	1	2	yes	—
297	C₂×C₄	C₆₇⋊#95	1	2	yes	—
298	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
299	C₂×C₄	#14523954	1	4	yes	—
300	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	4	yes	—
301	C₂×C₄	#14523963	1	4	yes	—
302	C₂×C₄	#14523964	1	2	yes	—
303	C₂×C₄	#14523966	1	4	yes	—
304	C₂×C₄	#14523967	1	2	yes	—
305	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	8	yes	—
306	C₂×C₄	#14523974	1	4	yes	—
307	C₂×C₄	#14523976	1	4	yes	—
308	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
312	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
315	C₂×C₄	C₂×#11761612	1	2	yes	—
316	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
317	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
318	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
319	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
320	C₈	—	268	4	no	—
321	C₈	—	134	8	no	—
322	C₂²	D₄⋊(C₆₇⋊C₈)	1	8	yes	—
323	C₂²	Q₈⋊(C₆₇⋊C₈)	1	8	yes	—
324	C₂²	#16863151	1	4	yes	—
325	C₂²	#16863155	1	4	yes	—
326	C₂²	#16863158	1	4	yes	—
327	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
328	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
329	C₂²	C₂×#14523964	1	1	yes	characteristic
330	C₂²	C₂×#14523967	1	1	yes	characteristic
331	C₂²	C₂×#14523970	1	2	yes	—
332	C₄	—	2	1	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	8	no	—
336	C₄	#16863151	1	4	yes	—
337	C₄	#16863158	1	4	yes	—
338	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
339	C₄	C₂×#14523954	1	1	yes	characteristic
340	C₄	C₂×#14523963	1	1	yes	characteristic
341	C₄	C₂×#14523966	1	1	yes	characteristic
342	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	2	yes	—
343	C₄	C₂×#14523974	1	1	yes	characteristic
344	C₄	C₂×#14523976	1	1	yes	characteristic
345	C₂	#22558050	1	8	yes	direct product factor
346	C₂	C₂×D₄⋊(C₆₇⋊C₈)	1	2	yes	—
347	C₂	C₂×Q₈⋊(C₆₇⋊C₈)	1	2	yes	—
348	C₂	C₂×#16863151	1	1	yes	characteristic
349	C₂	C₂×#16863155	1	1	yes	characteristic
350	C₂	C₂×#16863158	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	C₂×#1279
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #22558050	C₂	direct (N × H)
9. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
13. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
32. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
35. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
42. C₂×C₄⋊C₄×C₆₇	C₈	semidirect
91. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
170. C₂×C₆₇	#1279	semidirect
180. C₆₇	C₂×#1279	semidirect
212. C₂×C₄⋊C₄	C₆₇⋊C₈	semidirect
267. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
345. C₂	#22558050	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂²×C₄×C₆₇⋊C₈	2
#22558050	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#1279	67

Subgroups — C2×#22558050 #30745356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22558050 #30745356