Subgroups — C₂×C₄×#102×C₆₇ #30699657

Order 17152.

2158 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#102×C₆₇	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#102×C₆₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#340×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	—	2	1	no	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₆₇	—	2	2	no	—
11	C₂×C₄×D₄×C₆₇	—	2	4	no	—
12	C₂×C₄×D₄×C₆₇	C₄	1	8	yes	complemented
13	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
17	C₂²×C₄²×C₆₇	—	2	1	no	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#102×C₆₇	C₄	1	16	yes	direct product factor
20	C₂×#102×C₆₇	C₂²	1	8	yes	—
21	C₂×C₄×C₈×C₆₇	—	2	4	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	complemented
23	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#95×C₆₇	—	2	1	no	—
26	#340×C₆₇	C₂²	1	8	yes	complemented
27	C₂⁵×C₆₇	—	2	1	no	—
28	C₂²×D₄×C₆₇	—	2	1	no	—
29	C₂²×D₄×C₆₇	—	2	2	no	—
30	C₂²×D₄×C₆₇	—	2	4	no	—
31	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
33	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	—	2	2	no	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	—	2	8	no	—
37	C₂³×C₄×C₆₇	—	2	4	no	—
38	C₂³×C₄×C₆₇	—	4	4	no	—
39	C₂³×C₄×C₆₇	—	4	4	no	—
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	C₄×D₄×C₆₇	—	2	16	no	—
48	C₄×D₄×C₆₇	—	2	8	no	—
49	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
60	C₂×C₄²×C₆₇	—	2	4	no	—
61	C₂×C₄²×C₆₇	—	4	4	no	—
62	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₆₇	D₄	1	2	yes	—
64	C₂×C₄²×C₆₇	D₄	1	4	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	#102×C₆₇	C₂×C₄	1	32	yes	direct product factor
67	C₄×C₈×C₆₇	—	2	8	no	—
68	#95×C₆₇	—	2	4	no	—
69	C₂⁴×C₆₇	—	2	4	no	—
70	C₂⁴×C₆₇	—	2	8	no	—
71	C₂⁴×C₆₇	—	2	2	no	—
72	C₂⁴×C₆₇	—	4	2	no	—
73	C₂⁴×C₆₇	—	4	2	no	—
74	C₂⁴×C₆₇	—	4	4	no	—
75	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×D₄×C₆₇	—	2	4	no	—
77	C₂×D₄×C₆₇	—	2	8	no	—
78	C₂×D₄×C₆₇	—	2	16	no	—
79	C₂×D₄×C₆₇	—	2	8	no	—
80	C₂×D₄×C₆₇	—	2	2	no	—
81	C₂×D₄×C₆₇	—	2	4	no	—
82	C₂×D₄×C₆₇	C₄²	1	16	yes	complemented
83	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
84	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	8	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	16	no	—
91	C₂²×C₄×C₆₇	—	4	4	no	—
92	C₂²×C₄×C₆₇	—	4	2	no	—
93	C₂²×C₄×C₆₇	—	4	2	no	—
94	C₂²×C₄×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	—	4	32	no	—
96	C₂²×C₄×C₆₇	—	4	8	no	—
97	C₂²×C₄×C₆₇	—	4	8	no	—
98	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
100	C₂²×C₄×C₆₇	D₈	1	4	yes	—
101	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	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	#41	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₆₇	—	2	16	no	—
111	C₂×C₈×C₆₇	—	2	4	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	complemented
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	8	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	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	complemented
120	C₂²⋊C₄×C₆₇	—	2	16	no	—
121	C₄²×C₆₇	—	4	8	no	—
122	C₄²×C₆₇	C₂²⋊C₄	1	8	yes	—
123	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
124	C₄²×C₆₇	C₂×D₄	1	4	yes	—
125	C₂³×C₆₇	—	2	4	no	—
126	C₂³×C₆₇	—	2	8	no	—
127	C₂³×C₆₇	—	2	16	no	—
128	C₂³×C₆₇	—	4	2	no	—
129	C₂³×C₆₇	—	4	2	no	—
130	C₂³×C₆₇	—	4	4	no	—
131	C₂³×C₆₇	—	4	8	no	—
132	C₂³×C₆₇	—	4	8	no	—
133	C₂³×C₆₇	—	4	16	no	—
134	C₂³×C₆₇	—	4	16	no	—
135	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂³×C₆₇	C₄×D₄	1	4	yes	—
137	C₂³×C₆₇	C₂×D₈	1	2	yes	—
138	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
139	C₂³×C₆₇	#135	1	4	yes	—
140	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	D₄×C₆₇	—	2	4	no	—
142	D₄×C₆₇	—	2	8	no	—
143	D₄×C₆₇	—	2	16	no	—
144	D₄×C₆₇	C₂×C₄²	1	8	yes	complemented
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	8	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	8	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	4	8	no	—
154	C₂×C₄×C₆₇	—	4	16	no	—
155	C₂×C₄×C₆₇	—	4	32	no	—
156	C₂×C₄×C₆₇	#102	1	16	yes	direct product factor
157	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
158	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
159	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
161	C₂×C₄×C₆₇	#117	1	4	yes	—
162	C₂×C₄×C₆₇	C₄×D₄	1	8	yes	—
163	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
165	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
166	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
170	C₈×C₆₇	—	2	8	no	—
171	C₂²×C₆₇	—	2	8	no	—
172	C₂²×C₆₇	—	4	4	no	—
173	C₂²×C₆₇	—	4	4	no	—
174	C₂²×C₆₇	—	4	8	no	—
175	C₂²×C₆₇	—	4	8	no	—
176	C₂²×C₆₇	—	4	32	no	—
177	C₂²×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
178	C₂²×C₆₇	C₂×#102	1	4	yes	—
179	C₂²×C₆₇	#416	1	4	yes	—
180	C₂²×C₆₇	C₄×D₈	1	8	yes	—
181	C₂²×C₆₇	C₄×SD₁₆	1	8	yes	—
182	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
184	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
185	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
186	C₂²×C₆₇	C₂×#135	1	2	yes	—
187	C₄×C₆₇	—	2	4	no	—
188	C₄×C₆₇	—	4	4	no	—
189	C₄×C₆₇	—	4	8	no	—
190	C₄×C₆₇	C₄×C₂²⋊C₄	1	4	yes	—
191	C₄×C₆₇	C₂×#102	1	8	yes	direct product factor
192	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
193	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
194	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
195	C₂×C₄×#102	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	4	8	no	—
197	C₂×C₆₇	C₄×#102	1	8	yes	direct product factor
198	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
201	C₂×C₆₇	C₂×C₄×D₈	1	2	yes	—
202	C₂×C₆₇	C₂×C₄×SD₁₆	1	2	yes	—
203	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
204	C₂²×#102	C₂×C₆₇	1	2	yes	—
205	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
206	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
207	C₄×#102	C₂×C₆₇	1	8	yes	direct product factor
208	C₂×#340	C₂×C₆₇	1	2	yes	complemented
209	C₆₇	C₂×C₄×#102	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
211	C₂⁴×C₄	—	2	1	no	—
212	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
213	C₂×C₄×D₄	—	2	2	no	—
214	C₂×C₄×D₄	—	2	4	no	—
215	C₂×C₄×D₄	C₄×C₆₇	1	8	yes	complemented
216	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
217	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
218	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
219	C₂²×C₂²⋊C₄	—	2	1	no	—
220	C₂²×C₄²	—	2	1	no	—
221	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#102	C₄×C₆₇	1	16	yes	direct product factor
223	C₂×#102	C₂²×C₆₇	1	8	yes	—
224	C₂×C₄×C₈	—	2	4	no	—
225	C₂×C₄×C₈	C₂²×C₆₇	1	4	yes	complemented
226	C₄×C₄⋊C₄	—	2	2	no	—
227	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
228	C₂×#95	—	2	1	no	—
229	#340	C₂²×C₆₇	1	8	yes	complemented
230	C₂⁵	—	2	1	no	—
231	C₂²×D₄	—	2	1	no	—
232	C₂²×D₄	—	2	2	no	—
233	C₂²×D₄	—	2	4	no	—
234	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
235	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
236	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
237	C₂³×C₄	—	2	1	no	—
238	C₂³×C₄	—	2	2	no	—
239	C₂³×C₄	—	2	8	no	—
240	C₂³×C₄	—	2	4	no	—
241	C₂³×C₄	—	4	4	no	—
242	C₂³×C₄	—	4	4	no	—
243	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
244	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×C₈	—	2	4	no	—
247	C₂²×C₈	—	2	4	no	—
248	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
249	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
250	C₄×D₄	—	2	16	no	—
251	C₄×D₄	—	2	8	no	—
252	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	4	no	—
257	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
259	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
261	C₂×C₂²⋊C₄	—	2	8	no	—
262	C₂×C₂²⋊C₄	—	2	4	no	—
263	C₂×C₄²	—	2	4	no	—
264	C₂×C₄²	—	4	4	no	—
265	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
266	C₂×C₄²	D₄×C₆₇	1	2	yes	—
267	C₂×C₄²	D₄×C₆₇	1	4	yes	—
268	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
269	#102	C₂×C₄×C₆₇	1	32	yes	direct product factor
270	C₄×C₈	—	2	8	no	—
271	#95	—	2	4	no	—
272	C₂⁴	—	2	4	no	—
273	C₂⁴	—	2	8	no	—
274	C₂⁴	—	2	2	no	—
275	C₂⁴	—	4	2	no	—
276	C₂⁴	—	4	2	no	—
277	C₂⁴	—	4	4	no	—
278	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂×D₄	—	2	4	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	2	16	no	—
282	C₂×D₄	—	2	8	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	4	no	—
285	C₂×D₄	C₄²×C₆₇	1	16	yes	complemented
286	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
287	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	—	2	8	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	16	no	—
294	C₂²×C₄	—	4	4	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	4	no	—
298	C₂²×C₄	—	4	32	no	—
299	C₂²×C₄	—	4	8	no	—
300	C₂²×C₄	—	4	8	no	—
301	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
302	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
303	C₂²×C₄	D₈×C₆₇	1	4	yes	—
304	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
305	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
307	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
311	C₂²×C₄	#41×C₆₇	1	4	yes	—
312	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
313	C₂×C₈	—	2	16	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	2	4	no	—
316	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	complemented
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
322	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	complemented
323	C₂²⋊C₄	—	2	16	no	—
324	C₄²	—	4	8	no	—
325	C₄²	C₂²⋊C₄×C₆₇	1	8	yes	—
326	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
327	C₄²	C₂×D₄×C₆₇	1	4	yes	—
328	C₂³	—	2	4	no	—
329	C₂³	—	2	8	no	—
330	C₂³	—	2	16	no	—
331	C₂³	—	4	2	no	—
332	C₂³	—	4	2	no	—
333	C₂³	—	4	4	no	—
334	C₂³	—	4	8	no	—
335	C₂³	—	4	8	no	—
336	C₂³	—	4	16	no	—
337	C₂³	—	4	16	no	—
338	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
339	C₂³	C₄×D₄×C₆₇	1	4	yes	—
340	C₂³	C₂×D₈×C₆₇	1	2	yes	—
341	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
342	C₂³	#135×C₆₇	1	4	yes	—
343	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
344	D₄	—	2	4	no	—
345	D₄	—	2	8	no	—
346	D₄	—	2	16	no	—
347	D₄	C₂×C₄²×C₆₇	1	8	yes	complemented
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	8	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	8	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	8	no	—
357	C₂×C₄	—	4	16	no	—
358	C₂×C₄	—	4	32	no	—
359	C₂×C₄	#102×C₆₇	1	16	yes	direct product factor
360	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
361	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
362	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
363	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
364	C₂×C₄	#117×C₆₇	1	4	yes	—
365	C₂×C₄	C₄×D₄×C₆₇	1	8	yes	—
366	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
367	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
368	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
369	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
370	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
371	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
373	C₈	—	2	8	no	—
374	C₂²	—	2	8	no	—
375	C₂²	—	4	32	no	—
376	C₂²	—	4	4	no	—
377	C₂²	—	4	4	no	—
378	C₂²	—	4	8	no	—
379	C₂²	—	4	8	no	—
380	C₂²	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
381	C₂²	C₂×#102×C₆₇	1	4	yes	—
382	C₂²	#416×C₆₇	1	4	yes	—
383	C₂²	C₄×D₈×C₆₇	1	8	yes	—
384	C₂²	C₄×SD₁₆×C₆₇	1	8	yes	—
385	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
386	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
387	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
388	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
389	C₂²	C₂×#135×C₆₇	1	2	yes	—
390	C₄	—	2	4	no	—
391	C₄	—	4	4	no	—
392	C₄	—	4	8	no	—
393	C₄	C₄×C₂²⋊C₄×C₆₇	1	4	yes	—
394	C₄	C₂×#102×C₆₇	1	8	yes	direct product factor
395	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
396	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
397	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
398	C₂	—	4	8	no	—
399	C₂	C₄×#102×C₆₇	1	8	yes	direct product factor
400	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
401	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic
402	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
403	C₂	C₂×C₄×D₈×C₆₇	1	2	yes	—
404	C₂	C₂×C₄×SD₁₆×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₈×C₆₇	C₂	semidirect
4. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
5. C₄×#102×C₆₇	C₂	direct (N × H)
6. C₂×#340×C₆₇	C₂	semidirect
12. C₂×C₄×D₄×C₆₇	C₄	semidirect
19. C₂×#102×C₆₇	C₄	direct (N × H)
22. C₂×C₄×C₈×C₆₇	C₂²	semidirect
24. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
26. #340×C₆₇	C₂²	semidirect
45. C₂²×C₈×C₆₇	C₂×C₄	semidirect
49. C₄×D₄×C₆₇	C₂×C₄	semidirect
55. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
66. #102×C₆₇	C₂×C₄	direct (N × H)
82. C₂×D₄×C₆₇	C₄²	semidirect
113. C₂×C₈×C₆₇	C₂²×C₄	semidirect
119. C₄⋊C₄×C₆₇	C₂²×C₄	semidirect
144. D₄×C₆₇	C₂×C₄²	semidirect
156. C₂×C₄×C₆₇	#102	direct (N × H)
191. C₄×C₆₇	C₂×#102	direct (N × H)
195. C₂×C₄×#102	C₆₇	direct (N × H)
197. C₂×C₆₇	C₄×#102	direct (N × H)
205. C₂²×C₄×C₈	C₂×C₆₇	semidirect
206. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
207. C₄×#102	C₂×C₆₇	direct (N × H)
208. C₂×#340	C₂×C₆₇	semidirect
209. C₆₇	C₂×C₄×#102	direct (N × H)
215. C₂×C₄×D₄	C₄×C₆₇	semidirect
222. C₂×#102	C₄×C₆₇	direct (N × H)
225. C₂×C₄×C₈	C₂²×C₆₇	semidirect
227. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
229. #340	C₂²×C₆₇	semidirect
248. C₂²×C₈	C₂×C₄×C₆₇	semidirect
252. C₄×D₄	C₂×C₄×C₆₇	semidirect
258. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
269. #102	C₂×C₄×C₆₇	direct (N × H)
285. C₂×D₄	C₄²×C₆₇	semidirect
316. C₂×C₈	C₂²×C₄×C₆₇	semidirect
322. C₄⋊C₄	C₂²×C₄×C₆₇	semidirect
347. D₄	C₂×C₄²×C₆₇	semidirect
359. C₂×C₄	#102×C₆₇	direct (N × H)
394. C₄	C₂×#102×C₆₇	direct (N × H)
399. C₂	C₄×#102×C₆₇	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×D₄×C₆₇	2
C₂²×#102×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₄×#102×C₆₇	2
C₂×#340×C₆₇	2
C₂×C₄×#102	67

Subgroups — C2×C4×#102×C67 #30699657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#102×C₆₇ #30699657