Subgroups — C₂×C₆₇×#1548 #30696956

Order 17152.

1246 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1548	C₂	1	8	yes	maximal, direct product factor
5	C₂×#333×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂³×Q₈×C₆₇	C₄	1	2	yes	—
7	C₂²×M₁₆×C₆₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₄	1	4	yes	—
13	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#431×C₆₇	—	2	2	no	—
20	#431×C₆₇	C₂²	1	4	yes	—
21	#423×C₆₇	—	2	2	no	—
22	#423×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
27	C₂×C₄×C₈×C₆₇	—	2	1	no	—
28	#333×C₆₇	C₂²	1	16	yes	complemented
29	C₂²×Q₈×C₆₇	—	2	4	no	—
30	C₂²×Q₈×C₆₇	—	2	2	no	—
31	C₂²×Q₈×C₆₇	—	4	8	no	—
32	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₆₇	—	2	2	no	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₆₇	—	2	4	no	—
39	C₂×M₁₆×C₆₇	—	4	4	no	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	—	2	4	no	—
43	C₄⋊Q₈×C₆₇	—	2	4	no	—
44	C₄⋊Q₈×C₆₇	—	2	4	no	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	—	2	4	no	—
48	C₄⋊Q₈×C₆₇	—	2	4	no	—
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
51	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	D₄	1	2	yes	—
65	C₂×C₄²×C₆₇	D₄	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	—	2	4	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	2	no	—
71	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
72	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
73	C₂²⋊C₈×C₆₇	—	2	4	no	—
74	C₄×C₈×C₆₇	—	2	4	no	—
75	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	—	2	8	no	—
78	C₂×Q₈×C₆₇	—	4	16	no	—
79	C₂×Q₈×C₆₇	—	4	16	no	—
80	C₂×Q₈×C₆₇	—	4	8	no	—
81	C₂×Q₈×C₆₇	—	4	8	no	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	8	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	4	8	no	—
88	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
95	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	M₁₆×C₆₇	—	4	8	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	4	4	no	—
106	C₂×C₈×C₆₇	—	4	4	no	—
107	C₄⋊C₄×C₆₇	—	2	8	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
114	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
115	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
116	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₆₇	C₂×D₄	1	2	yes	—
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	#103	1	4	yes	—
126	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	Q₈×C₆₇	—	4	8	no	—
129	Q₈×C₆₇	—	4	8	no	—
130	Q₈×C₆₇	—	4	16	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	4	16	no	—
139	C₂×C₄×C₆₇	—	4	8	no	—
140	C₂×C₄×C₆₇	#100	1	4	yes	—
141	C₂×C₄×C₆₇	#103	1	8	yes	—
142	C₂×C₄×C₆₇	#103	1	4	yes	—
143	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
151	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
152	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
153	C₂×C₄×C₆₇	#137	1	4	yes	—
154	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₈×C₆₇	—	4	4	no	—
158	C₈×C₆₇	—	4	4	no	—
159	C₂²×C₆₇	—	2	8	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	C₂×#100	1	2	yes	—
163	C₂²×C₆₇	C₂×#103	1	4	yes	—
164	C₂²×C₆₇	C₂×#103	1	2	yes	—
165	C₂²×C₆₇	#419	1	2	yes	—
166	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	4	8	no	—
171	C₄×C₆₇	C₂×#100	1	4	yes	—
172	C₄×C₆₇	C₂×#103	1	2	yes	—
173	C₄×C₆₇	C₂×#103	1	4	yes	—
174	C₄×C₆₇	#419	1	2	yes	—
175	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
176	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
177	C₄×C₆₇	C₂×#137	1	2	yes	—
178	C₂×#1548	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂×C₆₇	—	2	4	no	—
180	C₂×C₆₇	#1548	1	4	yes	direct product factor
181	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
182	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#419	1	1	yes	characteristic
184	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
185	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
186	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
187	#1548	C₂×C₆₇	1	8	yes	direct product factor
188	C₂×#333	C₂×C₆₇	1	4	yes	complemented
189	C₆₇	C₂×#1548	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³×Q₈	C₄×C₆₇	1	2	yes	—
191	C₂²×M₁₆	—	2	1	no	—
192	C₂×C₄⋊Q₈	—	2	1	no	—
193	C₂×C₄⋊Q₈	—	2	1	no	—
194	C₂×C₄⋊Q₈	—	2	2	no	—
195	C₂×C₄⋊Q₈	—	2	2	no	—
196	C₂×C₄⋊Q₈	C₄×C₆₇	1	4	yes	—
197	C₂×C₄⋊Q₈	C₄×C₆₇	1	2	yes	—
198	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
199	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
200	C₂²×C₄⋊C₄	—	2	1	no	—
201	C₂²×C₄⋊C₄	—	2	1	no	—
202	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
203	#431	—	2	2	no	—
204	#431	C₂²×C₆₇	1	4	yes	—
205	#423	—	2	2	no	—
206	#423	C₂²×C₆₇	1	4	yes	—
207	C₂×C₄⋊C₈	—	2	1	no	—
208	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
209	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
210	C₂×C₂²⋊C₈	—	2	1	no	—
211	C₂×C₄×C₈	—	2	1	no	—
212	#333	C₂²×C₆₇	1	16	yes	complemented
213	C₂²×Q₈	—	2	4	no	—
214	C₂²×Q₈	—	2	2	no	—
215	C₂²×Q₈	—	4	8	no	—
216	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
217	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
218	C₂³×C₄	—	2	2	no	—
219	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
220	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
221	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
222	C₂×M₁₆	—	2	4	no	—
223	C₂×M₁₆	—	4	4	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	4	no	—
227	C₄⋊Q₈	—	2	4	no	—
228	C₄⋊Q₈	—	2	4	no	—
229	C₄⋊Q₈	—	2	4	no	—
230	C₄⋊Q₈	—	2	2	no	—
231	C₄⋊Q₈	—	2	2	no	—
232	C₄⋊Q₈	—	2	4	no	—
233	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
234	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
235	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	4	no	—
238	C₂×C₄⋊C₄	—	2	4	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	—	2	4	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	2	4	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
246	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
247	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄²	D₄×C₆₇	1	2	yes	—
249	C₂×C₄²	D₄×C₆₇	1	2	yes	—
250	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
256	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
257	C₂²⋊C₈	—	2	4	no	—
258	C₄×C₈	—	2	4	no	—
259	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	2	8	no	—
262	C₂×Q₈	—	4	16	no	—
263	C₂×Q₈	—	4	16	no	—
264	C₂×Q₈	—	4	8	no	—
265	C₂×Q₈	—	4	8	no	—
266	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	2	8	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	4	8	no	—
272	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
274	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
275	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
276	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
277	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
279	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
280	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
282	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
283	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
286	M₁₆	—	4	8	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	4	4	no	—
290	C₂×C₈	—	4	4	no	—
291	C₄⋊C₄	—	2	8	no	—
292	C₄⋊C₄	—	2	8	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
298	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
299	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
300	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
301	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₆₇	1	2	yes	—
303	C₄²	C₂×D₄×C₆₇	1	2	yes	—
304	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	#103×C₆₇	1	4	yes	—
310	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
311	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
312	Q₈	—	4	8	no	—
313	Q₈	—	4	8	no	—
314	Q₈	—	4	16	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	8	no	—
322	C₂×C₄	—	4	16	no	—
323	C₂×C₄	—	4	8	no	—
324	C₂×C₄	#100×C₆₇	1	4	yes	—
325	C₂×C₄	#103×C₆₇	1	8	yes	—
326	C₂×C₄	#103×C₆₇	1	4	yes	—
327	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
328	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
331	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
334	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
335	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
336	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
337	C₂×C₄	#137×C₆₇	1	4	yes	—
338	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	4	4	no	—
343	C₂²	—	2	8	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	C₂×#100×C₆₇	1	2	yes	—
347	C₂²	C₂×#103×C₆₇	1	4	yes	—
348	C₂²	C₂×#103×C₆₇	1	2	yes	—
349	C₂²	#419×C₆₇	1	2	yes	—
350	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	8	no	—
355	C₄	C₂×#100×C₆₇	1	4	yes	—
356	C₄	C₂×#103×C₆₇	1	2	yes	—
357	C₄	C₂×#103×C₆₇	1	4	yes	—
358	C₄	#419×C₆₇	1	2	yes	—
359	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
360	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
361	C₄	C₂×#137×C₆₇	1	2	yes	—
362	C₂	—	2	4	no	—
363	C₂	C₆₇×#1548	1	4	yes	direct product factor
364	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
365	C₂	C₂²×#103×C₆₇	1	1	yes	characteristic
366	C₂	C₂×#419×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1548	C₂	direct (N × H)
5. C₂×#333×C₆₇	C₂	semidirect
28. #333×C₆₇	C₂²	semidirect
178. C₂×#1548	C₆₇	direct (N × H)
180. C₂×C₆₇	#1548	direct (N × H)
187. #1548	C₂×C₆₇	direct (N × H)
188. C₂×#333	C₂×C₆₇	semidirect
189. C₆₇	C₂×#1548	direct (N × H)
212. #333	C₂²×C₆₇	semidirect
363. C₂	C₆₇×#1548	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⋊Q₈×C₆₇	2
C₂×#431×C₆₇	2
C₂×#423×C₆₇	2
C₆₇×#1548	2
C₂×#333×C₆₇	2
C₂×#1548	67

Subgroups — C2×C67×#1548 #30696956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1548 #30696956