Subgroups — C₂×C₆₇×#1552 #30696961

Order 17152.

1030 conjugacy classes of subgroups.

The subgroup lattice has 404 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×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1552	C₂	1	8	yes	maximal, direct product factor
5	C₂×#332×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#332×C₆₇	C₂	1	2	yes	maximal, complemented
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	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#125×C₆₇	C₄	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
19	C₂×#117×C₆₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#432×C₆₇	—	2	2	no	—
22	#432×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
24	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
28	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
29	C₂×#97×C₆₇	—	2	1	no	—
30	#332×C₆₇	C₂²	1	8	yes	complemented
31	#332×C₆₇	C₂²	1	8	yes	complemented
32	C₂²×Q₈×C₆₇	—	2	1	no	—
33	C₂²×Q₈×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	4	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	—	4	4	no	—
41	C₄⋊Q₈×C₆₇	—	2	1	no	—
42	C₄⋊Q₈×C₆₇	—	2	1	no	—
43	C₄⋊Q₈×C₆₇	—	2	1	no	—
44	C₄⋊Q₈×C₆₇	—	2	1	no	—
45	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
49	#125×C₆₇	—	2	4	no	—
50	#125×C₆₇	C₂×C₄	1	8	yes	—
51	C₂²⋊Q₈×C₆₇	—	2	4	no	—
52	C₂²⋊Q₈×C₆₇	—	2	4	no	—
53	C₂²⋊Q₈×C₆₇	—	2	4	no	—
54	C₂²⋊Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	—	2	4	no	—
57	C₄×Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	—	2	4	no	—
59	#117×C₆₇	—	2	4	no	—
60	#117×C₆₇	C₂×C₄	1	8	yes	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	D₄	1	2	yes	—
74	C₂×C₄²×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
81	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
82	C₂²⋊C₈×C₆₇	—	2	4	no	—
83	#97×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	4	4	no	—
90	C₂×Q₈×C₆₇	—	4	2	no	—
91	C₂×Q₈×C₆₇	—	4	2	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	4	8	no	—
113	C₂×C₈×C₆₇	—	4	4	no	—
114	C₂×C₈×C₆₇	—	4	4	no	—
115	C₄⋊C₄×C₆₇	—	2	8	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
124	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂²⋊C₄×C₆₇	—	2	8	no	—
126	C₄²×C₆₇	—	2	8	no	—
127	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
128	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
129	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
130	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
131	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₆₇	C₂×D₄	1	2	yes	—
133	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂×D₄	1	2	yes	—
136	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂³×C₆₇	—	2	2	no	—
138	C₂³×C₆₇	—	2	2	no	—
139	C₂³×C₆₇	#101	1	4	yes	—
140	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	Q₈×C₆₇	—	4	4	no	—
143	Q₈×C₆₇	—	4	2	no	—
144	Q₈×C₆₇	—	4	2	no	—
145	Q₈×C₆₇	—	4	4	no	—
146	Q₈×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	4	4	no	—
162	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₆₇	#135	1	4	yes	—
169	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
170	C₂×C₄×C₆₇	#137	1	2	yes	—
171	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₆₇	—	4	4	no	—
175	C₈×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	2	8	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	C₂×#101	1	4	yes	—
180	C₂²×C₆₇	C₂×#101	1	2	yes	—
181	C₂²×C₆₇	#416	1	2	yes	—
182	C₂²×C₆₇	#417	1	2	yes	—
183	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	1	no	—
186	C₄×C₆₇	—	2	1	no	—
187	C₄×C₆₇	—	2	2	no	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	4	4	no	—
190	C₄×C₆₇	—	4	4	no	—
191	C₄×C₆₇	#416	1	2	yes	—
192	C₄×C₆₇	#417	1	2	yes	—
193	C₄×C₆₇	C₂×#135	1	2	yes	—
194	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
196	C₂×#1552	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₆₇	—	2	4	no	—
198	C₂×C₆₇	#1552	1	4	yes	direct product factor
199	C₂×C₆₇	C₂²×#101	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
201	C₂×C₆₇	C₂×#417	1	1	yes	characteristic
202	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
203	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
204	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
205	#1552	C₂×C₆₇	1	8	yes	direct product factor
206	C₂×#332	C₂×C₆₇	1	2	yes	complemented
207	C₂×#332	C₂×C₆₇	1	2	yes	complemented
208	C₆₇	C₂×#1552	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³×C₈	—	2	1	no	—
210	#533	—	2	2	no	—
211	#533	—	2	2	no	—
212	#533	C₄×C₆₇	1	4	yes	—
213	#533	C₂²×C₆₇	1	4	yes	—
214	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
216	C₂×#125	C₄×C₆₇	1	2	yes	—
217	C₂×C₂²⋊Q₈	—	2	1	no	—
218	C₂×C₂²⋊Q₈	—	2	1	no	—
219	C₂×C₄×Q₈	—	2	1	no	—
220	C₂×C₄×Q₈	—	2	1	no	—
221	C₂×#117	C₄×C₆₇	1	2	yes	—
222	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
223	#432	—	2	2	no	—
224	#432	C₂²×C₆₇	1	4	yes	—
225	C₂×C₄⋊C₈	—	2	1	no	—
226	C₂×C₄⋊C₈	—	2	2	no	—
227	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
230	C₂×C₂²⋊C₈	—	2	1	no	—
231	C₂×#97	—	2	1	no	—
232	#332	C₂²×C₆₇	1	8	yes	complemented
233	#332	C₂²×C₆₇	1	8	yes	complemented
234	C₂²×Q₈	—	2	1	no	—
235	C₂²×Q₈	—	2	1	no	—
236	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
237	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
238	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₈	—	2	4	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	—	4	4	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
248	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
249	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
250	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
251	#125	—	2	4	no	—
252	#125	C₂×C₄×C₆₇	1	8	yes	—
253	C₂²⋊Q₈	—	2	4	no	—
254	C₂²⋊Q₈	—	2	4	no	—
255	C₂²⋊Q₈	—	2	4	no	—
256	C₂²⋊Q₈	—	2	4	no	—
257	C₄×Q₈	—	2	4	no	—
258	C₄×Q₈	—	2	4	no	—
259	C₄×Q₈	—	2	4	no	—
260	C₄×Q₈	—	2	4	no	—
261	#117	—	2	4	no	—
262	#117	C₂×C₄×C₆₇	1	8	yes	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
272	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	D₄×C₆₇	1	2	yes	—
276	C₂×C₄²	D₄×C₆₇	1	2	yes	—
277	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
279	C₄⋊C₈	—	2	4	no	—
280	C₄⋊C₈	—	2	2	no	—
281	C₄⋊C₈	—	2	2	no	—
282	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
283	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
284	C₂²⋊C₈	—	2	4	no	—
285	#97	—	2	4	no	—
286	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	4	4	no	—
292	C₂×Q₈	—	4	2	no	—
293	C₂×Q₈	—	4	2	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
303	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₂×C₈	—	2	4	no	—
313	C₂×C₈	—	2	4	no	—
314	C₂×C₈	—	4	8	no	—
315	C₂×C₈	—	4	4	no	—
316	C₂×C₈	—	4	4	no	—
317	C₄⋊C₄	—	2	8	no	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
326	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
327	C₂²⋊C₄	—	2	8	no	—
328	C₄²	—	2	8	no	—
329	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
330	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
331	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
332	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
333	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
334	C₄²	C₂×D₄×C₆₇	1	2	yes	—
335	C₄²	C₂×D₄×C₆₇	1	2	yes	—
336	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
339	C₂³	—	2	2	no	—
340	C₂³	—	2	2	no	—
341	C₂³	#101×C₆₇	1	4	yes	—
342	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
343	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
344	Q₈	—	4	4	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	4	2	no	—
347	Q₈	—	4	4	no	—
348	Q₈	—	4	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
365	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
367	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
370	C₂×C₄	#135×C₆₇	1	4	yes	—
371	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
372	C₂×C₄	#137×C₆₇	1	2	yes	—
373	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
376	C₈	—	4	4	no	—
377	C₈	—	4	4	no	—
378	C₂²	—	2	8	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	C₂×#101×C₆₇	1	4	yes	—
382	C₂²	C₂×#101×C₆₇	1	2	yes	—
383	C₂²	#416×C₆₇	1	2	yes	—
384	C₂²	#417×C₆₇	1	2	yes	—
385	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
386	C₄	—	2	2	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	4	no	—
392	C₄	—	4	4	no	—
393	C₄	#416×C₆₇	1	2	yes	—
394	C₄	#417×C₆₇	1	2	yes	—
395	C₄	C₂×#135×C₆₇	1	2	yes	—
396	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
397	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
398	C₂	—	2	4	no	—
399	C₂	C₆₇×#1552	1	4	yes	direct product factor
400	C₂	C₂²×#101×C₆₇	1	1	yes	characteristic
401	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
402	C₂	C₂×#417×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×#1552	C₂	direct (N × H)
5. C₂×#332×C₆₇	C₂	semidirect
6. C₂×#332×C₆₇	C₂	semidirect
30. #332×C₆₇	C₂²	semidirect
31. #332×C₆₇	C₂²	semidirect
196. C₂×#1552	C₆₇	direct (N × H)
198. C₂×C₆₇	#1552	direct (N × H)
205. #1552	C₂×C₆₇	direct (N × H)
206. C₂×#332	C₂×C₆₇	semidirect
207. C₂×#332	C₂×C₆₇	semidirect
208. C₆₇	C₂×#1552	direct (N × H)
232. #332	C₂²×C₆₇	semidirect
233. #332	C₂²×C₆₇	semidirect
399. C₂	C₆₇×#1552	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×#432×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₆₇×#1552	2
C₂×#332×C₆₇	2
C₂×#1552	67

Subgroups — C2×C67×#1552 #30696961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1552 #30696961