Subgroups — C₆₇×#20734 #30699985

Order 17152.

1370 conjugacy classes of subgroups.

The subgroup lattice has 386 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	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1824	C₂	1	2	yes	maximal
5	C₆₇×#1803	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1787	C₂	1	2	yes	maximal
7	C₄×#103×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₆₇	—	2	4	no	—
10	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
11	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
12	C₂×C₄×Q₈×C₆₇	C₄	1	8	yes	complemented
13	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	—	2	1	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#431×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	8	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	#384×C₆₇	C₂²	1	2	yes	—
26	#382×C₆₇	—	2	1	no	—
27	#380×C₆₇	—	2	1	no	—
28	#380×C₆₇	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₆₇	—	2	8	no	—
33	C₂²×Q₈×C₆₇	—	2	2	no	—
34	C₂²×Q₈×C₆₇	—	2	1	no	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₇	—	2	1	no	—
38	C₂³×C₄×C₆₇	—	2	1	no	—
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₄×Q₈×C₆₇	—	2	8	no	—
44	C₄×Q₈×C₆₇	—	2	8	no	—
45	C₄×Q₈×C₆₇	—	2	4	no	—
46	C₄×Q₈×C₆₇	—	2	4	no	—
47	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
48	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
49	#117×C₆₇	—	4	4	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	—	2	8	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	D₄	1	4	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	—	2	4	no	—
72	#103×C₆₇	—	2	16	no	—
73	C₂²⋊C₈×C₆₇	—	2	4	no	—
74	C₄×C₈×C₆₇	—	2	4	no	—
75	#95×C₆₇	—	2	2	no	—
76	#95×C₆₇	—	2	2	no	—
77	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×Q₈×C₆₇	—	2	16	no	—
79	C₂×Q₈×C₆₇	—	2	8	no	—
80	C₂×Q₈×C₆₇	—	2	16	no	—
81	C₂×Q₈×C₆₇	—	2	4	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂×Q₈×C₆₇	#41	1	8	yes	—
86	C₂²×C₄×C₆₇	—	2	8	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₆₇	—	2	4	no	—
108	C₂×C₈×C₆₇	—	2	8	no	—
109	C₄⋊C₄×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	2	8	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	8	no	—
114	C₄⋊C₄×C₆₇	—	4	4	no	—
115	C₄⋊C₄×C₆₇	—	4	4	no	—
116	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
117	C₂²⋊C₄×C₆₇	—	4	4	no	—
118	C₄²×C₆₇	—	2	8	no	—
119	C₄²×C₆₇	—	2	8	no	—
120	C₄²×C₆₇	—	4	4	no	—
121	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
122	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
123	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
124	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₄²×C₆₇	C₂×D₄	1	2	yes	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
131	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
132	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	Q₈×C₆₇	—	2	16	no	—
134	Q₈×C₆₇	—	2	4	no	—
135	Q₈×C₆₇	—	2	8	no	—
136	Q₈×C₆₇	#117	1	8	yes	complemented
137	C₂×C₄×C₆₇	—	2	16	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	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	—	4	4	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	4	no	—
151	C₂×C₄×C₆₇	—	4	4	no	—
152	C₂×C₄×C₆₇	#103	1	16	yes	complemented
153	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
161	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
162	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
165	C₈×C₆₇	—	4	4	no	—
166	C₂²×C₆₇	—	2	4	no	—
167	C₂²×C₆₇	—	2	4	no	—
168	C₂²×C₆₇	—	2	2	no	—
169	C₂²×C₆₇	—	2	2	no	—
170	C₂²×C₆₇	C₂×#103	1	4	yes	—
171	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₆₇	#525	1	2	yes	—
173	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
174	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
175	C₂²×C₆₇	#575	1	2	yes	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	8	no	—
179	C₄×C₆₇	—	2	2	no	—
180	C₄×C₆₇	—	4	4	no	—
181	C₄×C₆₇	—	8	4	no	—
182	C₄×C₆₇	C₂×#103	1	4	yes	—
183	C₄×C₆₇	C₂×#103	1	4	yes	—
184	C₄×C₆₇	C₂×#117	1	2	yes	—
185	C₄×C₆₇	#525	1	2	yes	—
186	#20734	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂×C₆₇	—	2	4	no	—
188	C₂×C₆₇	#2280	1	1	yes	characteristic
189	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
190	C₂×C₆₇	#2896	1	1	yes	characteristic
191	C₂×C₆₇	#3103	1	2	yes	—
192	C₂×C₆₇	#3104	1	2	yes	—
193	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
194	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
195	#2285	C₂×C₆₇	1	1	yes	characteristic
196	#1824	C₂×C₆₇	1	2	yes	—
197	#1803	C₂×C₆₇	1	4	yes	complemented
198	#1787	C₂×C₆₇	1	2	yes	—
199	C₄×#103	C₂×C₆₇	1	4	yes	complemented
200	C₆₇	#20734	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
202	C₂×C₄×Q₈	—	2	4	no	—
203	C₂×C₄×Q₈	—	2	1	no	—
204	C₂×C₄×Q₈	—	2	1	no	—
205	C₂×C₄×Q₈	C₄×C₆₇	1	8	yes	complemented
206	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
207	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
208	C₂×#117	—	2	1	no	—
209	C₂²×C₄⋊C₄	—	2	1	no	—
210	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
211	C₂²×C₄²	—	2	1	no	—
212	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
213	#431	—	2	4	no	—
214	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
215	C₂×#103	C₂²×C₆₇	1	8	yes	—
216	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
218	#384	C₂²×C₆₇	1	2	yes	—
219	#382	—	2	1	no	—
220	#380	—	2	1	no	—
221	#380	C₂²×C₆₇	1	2	yes	—
222	C₄×C₄⋊C₄	—	2	1	no	—
223	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
224	#340	C₂²×C₆₇	1	8	yes	—
225	C₂²×Q₈	—	2	8	no	—
226	C₂²×Q₈	—	2	2	no	—
227	C₂²×Q₈	—	2	1	no	—
228	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
229	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
230	C₂³×C₄	—	2	1	no	—
231	C₂³×C₄	—	2	1	no	—
232	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
233	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
234	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
235	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
236	C₄×Q₈	—	2	8	no	—
237	C₄×Q₈	—	2	8	no	—
238	C₄×Q₈	—	2	4	no	—
239	C₄×Q₈	—	2	4	no	—
240	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
241	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
242	#117	—	4	4	no	—
243	C₂×C₄⋊C₄	—	2	8	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	2	no	—
248	C₂×C₄⋊C₄	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
251	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
252	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
253	C₂×C₂²⋊C₄	—	2	2	no	—
254	C₂×C₄²	—	2	8	no	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	—	2	2	no	—
257	C₂×C₄²	—	2	2	no	—
258	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
259	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
260	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₄²	D₄×C₆₇	1	4	yes	—
262	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
264	C₄⋊C₈	—	2	4	no	—
265	#103	—	2	16	no	—
266	C₂²⋊C₈	—	2	4	no	—
267	C₄×C₈	—	2	4	no	—
268	#95	—	2	2	no	—
269	#95	—	2	2	no	—
270	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂×Q₈	—	2	16	no	—
272	C₂×Q₈	—	2	8	no	—
273	C₂×Q₈	—	2	16	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	4	no	—
277	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
278	C₂×Q₈	#41×C₆₇	1	8	yes	—
279	C₂²×C₄	—	2	8	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	4	4	no	—
288	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
289	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
290	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
291	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
292	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
293	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
297	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
299	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	2	8	no	—
302	C₄⋊C₄	—	2	8	no	—
303	C₄⋊C₄	—	2	8	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	8	no	—
307	C₄⋊C₄	—	4	4	no	—
308	C₄⋊C₄	—	4	4	no	—
309	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
310	C₂²⋊C₄	—	4	4	no	—
311	C₄²	—	2	8	no	—
312	C₄²	—	2	8	no	—
313	C₄²	—	4	4	no	—
314	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
315	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
316	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
317	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
318	C₄²	C₂×D₄×C₆₇	1	2	yes	—
319	C₄²	C₂×D₄×C₆₇	1	2	yes	—
320	C₂³	—	2	2	no	—
321	C₂³	—	2	2	no	—
322	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
323	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
324	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
325	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
326	Q₈	—	2	16	no	—
327	Q₈	—	2	4	no	—
328	Q₈	—	2	8	no	—
329	Q₈	#117×C₆₇	1	8	yes	complemented
330	C₂×C₄	—	2	16	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	8	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	8	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	#103×C₆₇	1	16	yes	complemented
346	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
347	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
348	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
349	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
350	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
351	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
352	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
353	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
354	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
355	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
358	C₈	—	4	4	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	4	no	—
361	C₂²	—	2	4	no	—
362	C₂²	—	2	2	no	—
363	C₂²	C₂×#103×C₆₇	1	4	yes	—
364	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
365	C₂²	#525×C₆₇	1	2	yes	—
366	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
367	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
368	C₂²	#575×C₆₇	1	2	yes	—
369	C₄	—	2	8	no	—
370	C₄	—	2	2	no	—
371	C₄	—	2	2	no	—
372	C₄	—	2	2	no	—
373	C₄	—	4	4	no	—
374	C₄	—	8	4	no	—
375	C₄	C₂×#103×C₆₇	1	4	yes	—
376	C₄	C₂×#103×C₆₇	1	4	yes	—
377	C₄	C₂×#117×C₆₇	1	2	yes	—
378	C₄	#525×C₆₇	1	2	yes	—
379	C₂	—	2	4	no	—
380	C₂	C₆₇×#2280	1	1	yes	characteristic
381	C₂	C₂²×#103×C₆₇	1	1	yes	characteristic
382	C₂	C₆₇×#2896	1	1	yes	characteristic
383	C₂	C₆₇×#3103	1	2	yes	—
384	C₂	C₆₇×#3104	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
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
5. C₆₇×#1803	C₂	semidirect
7. C₄×#103×C₆₇	C₂	semidirect
12. C₂×C₄×Q₈×C₆₇	C₄	semidirect
136. Q₈×C₆₇	#117	semidirect
152. C₂×C₄×C₆₇	#103	semidirect
186. #20734	C₆₇	direct (N × H)
197. #1803	C₂×C₆₇	semidirect
199. C₄×#103	C₂×C₆₇	semidirect
200. C₆₇	#20734	direct (N × H)
205. C₂×C₄×Q₈	C₄×C₆₇	semidirect
329. Q₈	#117×C₆₇	semidirect
345. C₂×C₄	#103×C₆₇	semidirect

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
C₆₇×#2285	2
C₆₇×#1824	2
C₆₇×#1803	2
C₆₇×#1787	2
C₄×#103×C₆₇	2
#20734	67

Subgroups — C67×#20734 #30699985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20734 #30699985