Subgroups — C₂×C₆₇×#1493 #30696722

Order 17152.

1150 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1493	C₂	1	8	yes	maximal, direct product factor
5	C₂×Q₈⋊C₈×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₆₇	—	2	1	no	—
7	C₂³×C₈×C₆₇	—	2	1	no	—
8	#533×C₆₇	—	2	2	no	—
9	#533×C₆₇	—	2	2	no	—
10	#533×C₆₇	C₄	1	4	yes	—
11	#533×C₆₇	C₂²	1	4	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
14	C₂×#125×C₆₇	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
17	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#117×C₆₇	—	2	1	no	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	—	2	2	no	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
29	Q₈⋊C₈×C₆₇	C₂²	1	16	yes	complemented
30	C₂²×Q₈×C₆₇	—	2	1	no	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	4	no	—
36	C₂×M₁₆×C₆₇	—	4	4	no	—
37	C₂²×C₈×C₆₇	—	2	8	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
45	#125×C₆₇	—	2	4	no	—
46	#125×C₆₇	—	2	4	no	—
47	C₂²⋊Q₈×C₆₇	—	2	4	no	—
48	C₂²⋊Q₈×C₆₇	—	2	4	no	—
49	C₂²⋊Q₈×C₆₇	—	2	4	no	—
50	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	2	2	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
57	#117×C₆₇	—	2	4	no	—
58	#117×C₆₇	—	2	4	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	D₄	1	2	yes	—
71	C₂×C₄²×C₆₇	D₄	1	2	yes	—
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	—	2	4	no	—
80	C₄×C₈×C₆₇	C₂³	1	4	yes	—
81	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	2	8	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
98	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	M₁₆×C₆₇	—	4	8	no	—
106	C₂×C₈×C₆₇	—	2	8	no	—
107	C₂×C₈×C₆₇	—	2	16	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₂×C₈×C₆₇	—	4	4	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₄⋊C₄×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	8	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂²⋊C₄×C₆₇	—	2	8	no	—
120	C₄²×C₆₇	—	2	4	no	—
121	C₄²×C₆₇	—	2	4	no	—
122	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
123	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
124	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
125	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₆₇	C₂×D₄	1	2	yes	—
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	#103	1	4	yes	—
135	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	Q₈×C₆₇	—	2	8	no	—
138	Q₈×C₆₇	—	4	4	no	—
139	Q₈×C₆₇	—	4	8	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	2	4	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	#103	1	8	yes	—
151	C₂×C₄×C₆₇	#103	1	4	yes	—
152	C₂×C₄×C₆₇	#104	1	4	yes	—
153	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×C₄×C₆₇	#131	1	4	yes	—
160	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
162	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
163	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
164	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₆₇	—	2	8	no	—
168	C₈×C₆₇	—	4	4	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	8	no	—
172	C₂²×C₆₇	#408	1	2	yes	—
173	C₂²×C₆₇	C₂×#103	1	4	yes	—
174	C₂²×C₆₇	C₂×#103	1	2	yes	—
175	C₂²×C₆₇	C₂×#104	1	2	yes	—
176	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	4	4	no	—
181	C₄×C₆₇	—	4	4	no	—
182	C₄×C₆₇	#408	1	2	yes	—
183	C₄×C₆₇	C₂×#103	1	2	yes	—
184	C₄×C₆₇	C₂×#103	1	4	yes	—
185	C₄×C₆₇	C₂×#104	1	4	yes	—
186	C₄×C₆₇	C₂×#131	1	2	yes	—
187	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
188	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
189	C₂×#1493	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₆₇	—	2	4	no	—
191	C₂×C₆₇	#1493	1	4	yes	direct product factor
192	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
193	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
194	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
195	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
196	C₂×#423	C₂×C₆₇	1	1	yes	characteristic
197	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
198	#1493	C₂×C₆₇	1	8	yes	direct product factor
199	C₂×Q₈⋊C₈	C₂×C₆₇	1	4	yes	complemented
200	C₆₇	C₂×#1493	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×M₁₆	—	2	1	no	—
202	C₂³×C₈	—	2	1	no	—
203	#533	—	2	2	no	—
204	#533	—	2	2	no	—
205	#533	C₄×C₆₇	1	4	yes	—
206	#533	C₂²×C₆₇	1	4	yes	—
207	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
209	C₂×#125	—	2	1	no	—
210	C₂×C₂²⋊Q₈	—	2	1	no	—
211	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
212	C₂×C₄×Q₈	—	2	1	no	—
213	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
214	C₂×#117	—	2	1	no	—
215	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
216	#423	—	2	2	no	—
217	#423	C₂²×C₆₇	1	4	yes	—
218	C₂×C₄⋊C₈	—	2	1	no	—
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄×C₈	—	2	1	no	—
221	C₂×C₄×C₈	—	2	2	no	—
222	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
223	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	—
224	Q₈⋊C₈	C₂²×C₆₇	1	16	yes	complemented
225	C₂²×Q₈	—	2	1	no	—
226	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
227	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
228	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
229	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
230	C₂×M₁₆	—	2	4	no	—
231	C₂×M₁₆	—	4	4	no	—
232	C₂²×C₈	—	2	8	no	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	—	2	4	no	—
235	C₂²×C₈	—	2	2	no	—
236	C₄⋊Q₈	—	2	2	no	—
237	C₄⋊Q₈	—	2	2	no	—
238	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
239	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
240	#125	—	2	4	no	—
241	#125	—	2	4	no	—
242	C₂²⋊Q₈	—	2	4	no	—
243	C₂²⋊Q₈	—	2	4	no	—
244	C₂²⋊Q₈	—	2	4	no	—
245	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
246	C₄×Q₈	—	2	4	no	—
247	C₄×Q₈	—	2	4	no	—
248	C₄×Q₈	—	2	2	no	—
249	C₄×Q₈	—	2	2	no	—
250	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
251	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
252	#117	—	2	4	no	—
253	#117	—	2	4	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₄²	—	2	2	no	—
262	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
264	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	D₄×C₆₇	1	2	yes	—
267	C₂×C₄²	D₄×C₆₇	1	2	yes	—
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
270	C₄⋊C₈	—	2	4	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
273	C₄×C₈	—	2	2	no	—
274	C₄×C₈	—	2	4	no	—
275	C₄×C₈	C₂³×C₆₇	1	4	yes	—
276	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	4	no	—
279	C₂×Q₈	—	2	8	no	—
280	C₂×Q₈	—	4	4	no	—
281	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
287	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
289	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
293	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
294	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
300	M₁₆	—	4	8	no	—
301	C₂×C₈	—	2	8	no	—
302	C₂×C₈	—	2	16	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	4	no	—
305	C₂×C₈	—	4	4	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	4	no	—
308	C₄⋊C₄	—	2	4	no	—
309	C₄⋊C₄	—	2	8	no	—
310	C₄⋊C₄	—	2	4	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
314	C₂²⋊C₄	—	2	8	no	—
315	C₄²	—	2	4	no	—
316	C₄²	—	2	4	no	—
317	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
318	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
319	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
320	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
321	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
322	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₄²	C₂×D₄×C₆₇	1	2	yes	—
324	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₄²	C₂×D₄×C₆₇	1	2	yes	—
326	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
327	C₂³	—	2	2	no	—
328	C₂³	—	2	2	no	—
329	C₂³	#103×C₆₇	1	4	yes	—
330	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
331	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
332	Q₈	—	2	8	no	—
333	Q₈	—	4	4	no	—
334	Q₈	—	4	8	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	#103×C₆₇	1	8	yes	—
346	C₂×C₄	#103×C₆₇	1	4	yes	—
347	C₂×C₄	#104×C₆₇	1	4	yes	—
348	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
350	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
353	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
354	C₂×C₄	#131×C₆₇	1	4	yes	—
355	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
356	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
357	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
358	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
359	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
361	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	C₈	—	2	8	no	—
363	C₈	—	4	4	no	—
364	C₂²	—	2	8	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	#408×C₆₇	1	2	yes	—
368	C₂²	C₂×#103×C₆₇	1	4	yes	—
369	C₂²	C₂×#103×C₆₇	1	2	yes	—
370	C₂²	C₂×#104×C₆₇	1	2	yes	—
371	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	4	no	—
376	C₄	—	4	4	no	—
377	C₄	#408×C₆₇	1	2	yes	—
378	C₄	C₂×#103×C₆₇	1	2	yes	—
379	C₄	C₂×#103×C₆₇	1	4	yes	—
380	C₄	C₂×#104×C₆₇	1	4	yes	—
381	C₄	C₂×#131×C₆₇	1	2	yes	—
382	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
383	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
384	C₂	—	2	4	no	—
385	C₂	C₆₇×#1493	1	4	yes	direct product factor
386	C₂	C₂×#408×C₆₇	1	1	yes	characteristic
387	C₂	C₂²×#103×C₆₇	1	1	yes	characteristic
388	C₂	C₂²×#104×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1493	C₂	direct (N × H)
5. C₂×Q₈⋊C₈×C₆₇	C₂	semidirect
29. Q₈⋊C₈×C₆₇	C₂²	semidirect
189. C₂×#1493	C₆₇	direct (N × H)
191. C₂×C₆₇	#1493	direct (N × H)
198. #1493	C₂×C₆₇	direct (N × H)
199. C₂×Q₈⋊C₈	C₂×C₆₇	semidirect
200. C₆₇	C₂×#1493	direct (N × H)
224. Q₈⋊C₈	C₂²×C₆₇	semidirect
385. C₂	C₆₇×#1493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂×#423×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#1493	2
C₂×Q₈⋊C₈×C₆₇	2
C₂×#1493	67

Subgroups — C2×C67×#1493 #30696722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1493 #30696722